

		SMALL SALEMS

		List of all Salem numbers of degree <= 20 and size < 1.3
		together with notes on K3 realizability
		Only 10.9 is missing from this list (it is about 1.43...)

X:  Not realizable, except perhaps with no twisting.
A:  Untwisted form is definitely obstructed
OK: Realizable

10.1 Salem value 1.1762808182599175065 OK A
                    3    4    5    6    7    9    10
Polynomial 1 + x - x  - x  - x  - x  - x  + x  + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, 1}
Feasible primes: {3, 5, 13, 23, 29}
Ideal 23         Periods {22}           Min 4  REALIZED
Ideal 3 13       Periods {4, 14}        Min 4  REALIZED

18.1 Salem value 1.1883681475082235881 OK A
                    2    3    6    7    8    9    10    11    12    15
Polynomial 1 - x + x  - x  - x  + x  - x  + x  - x   + x   - x   - x   + 
 
      16    17    18
>    x   - x   + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, 1}
Feasible primes: {7, 13}
Twists found: 1
Ideal 13         Periods {12}           Min 4  REALIZED

14.1 Salem value 1.2000265239873915189 X A
                3    4    7    10    11    14
Polynomial 1 - x  - x  + x  - x   - x   + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, 1}
Feasible primes: {5, 13, 19, 41}
Twists found: 3
Ideal 5          Periods {6}            Min 2  OBSTRUCTED
Ideal 13         Periods {7}            Min 2  OBSTRUCTED
Ideal 5 19       Periods {6, 9}         Min 4  Impossible
CANNOT BE REALIZED

14.2 Salem value 1.2026167436886042611 X A
                2    7    12    14
Polynomial 1 - x  - x  - x   + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, 1}
Feasible primes: {7, 11, 17, 29}
Twists found: 3
Ideal 7          Periods {8}            Min 2  OBSTRUCTED
Ideal 17         Periods {18}           Min 2  OBSTRUCTED
Ideal 7 11       Periods {5, 8}         Min 4  Impossible
CANNOT BE REALIZED

10.2 Salem value 1.2163916611382650916 OK A
                4    5    6    10
Polynomial 1 - x  - x  - x  + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, 1}
Feasible primes: {7, 11, 13, 17, 43}
Twists found: 9
Ideal 43         Periods {42}           Min 4  REALIZED

18.2 Salem value 1.2197208590403118442 X A
                    8    9    10    17    18
Polynomial 1 - x - x  + x  - x   - x   + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, 1}
Feasible primes: {5}
Twists found: 0
NOT REALIZABLE

10.3 Salem value 1.2303914344072247028 OK A
                3    5    7    10
Polynomial 1 - x  - x  - x  + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, 5}
Feasible primes: {5, 17, 23, 29, 83}
Twists found: 5
Ideal 17         Periods {2, 16}        Min 4  REALIZED

20.1 Salem value 1.2326135485931210040 X A
                    5    6    9    10    11    14    15    19    20
Polynomial 1 - x - x  + x  - x  + x   - x   + x   - x   - x   + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, -11}
Feasible primes: {11}
Twists found: 0
NOT REALIZABLE

16.1 Salem value 1.2363179318032304899 X A
                    8    15    16
Polynomial 1 - x - x  - x   + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, -3}
Feasible primes: {3, 29}
Twists found: 1
Ideal 3          Periods {2, 6}         Min 2  OBSTRUCTED
OBSTRUCTED FOR ALL TWISTS
NOT REALIZABLE

12.1 Salem value 1.2407264236525413921 ? A
                    2    3    6    9    10    11    12
Polynomial 1 - x + x  - x  - x  - x  + x   - x   + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, -7}
Feasible primes: {7, 13, 31}
Twists found: 3
Ideal 7          Periods {2, 14}        Min 4  
Ideal 13         Periods {2, 12}        Min 4  
Ideal 31         Periods {2, 30}        Min 4  
UNRESOLVED

18.3 Salem value 1.2527759374101139009 X A
                6    7    8    9    10    11    12    18
Polynomial 1 - x  - x  - x  - x  - x   - x   - x   + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, 5}
Feasible primes: {5}
Twists found: 0
NOT REALIZABLE

20.2 Salem value 1.2533306502014897570 X A
                2    5    15    18    20
Polynomial 1 - x  - x  - x   - x   + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, -4}
Feasible primes: {2}
Twists found: 1
Ideal 1		 Periods {4}	        Min 2 REALIZED?
Ideal 2          Periods {4}            Min 2 Not realizable

14.3 Salem value 1.2550935167637228792 ? A
                2    3    5    7    9    11    12    14
Polynomial 1 - x  - x  + x  - x  + x  - x   - x   + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, 1}
Feasible primes: {5, 7, 29, 37}
Twists found: 3
Ideal 7          Periods {8}            Min 2  OBSTRUCTED
Ideal 29         Periods {30}           Min 2  OBSTRUCTED
Ideal 5 7        Periods {4, 8}         Min 4  

18.4 Salem value 1.2562211543916702331 ?
                    4    5    9    13    14    17    18
Polynomial 1 - x - x  + x  - x  + x   - x   - x   + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, 1}
Feasible primes: {2, 17}
Twists found: 2
Ideal 17         Periods {8}            Min 4  Infeasible - needs more ramification
Ideal 2^2        Periods {6}            Min 2  

10.4 Salem value 1.2612309611371388519 OK
                2    5    8    10
Polynomial 1 - x  - x  - x  + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, 1}
Feasible primes: {2, 11, 13, 17, 53}
Twists found: 12
Ideal 2 13       Periods {3, 14}        Min 4  REALIZABLE

14.4 Salem value 1.2672964425230686927 ? A
                    6    7    8    13    14
Polynomial 1 - x - x  + x  - x  - x   + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, 1}
Feasible primes: {3, 5, 13, 17, 19}
Twists found: 4
Ideal 3          Periods {5}            Min 2  
Ideal 3          Periods {4}            Min 2  OBSTRUCTED
Ideal 13         Periods {7}            Min 2  OBSTRUCTED
Ideal 3 19       Periods {4, 18}        Min 4  

8.1 Salem value 1.2806381562677575967 OK A
                3    4    5    8
Polynomial 1 - x  - x  - x  + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, -3}
Feasible primes: {3, 7, 19, 23, 31, 43}
Twists found: 15
Ideal 23         Periods {2, 11}        Min 4  REALIZED
Ideal 3^2        Periods {2, 6, 18}     Min 4  REALIZED

20.3 Salem value 1.2824955606399601696 X - A
                        2      3      4      5    6    8    9    10    11
Polynomial 1 - 2 x + 2 x  - 2 x  + 2 x  - 2 x  + x  - x  + x  - x   + x   - 
 
      12    14      15      16      17      18      19    20
>    x   + x   - 2 x   + 2 x   - 2 x   + 2 x   - 2 x   + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, -19}
Feasible primes: {19}
Twists found: 0
NOT REALIZABLE

18.5 Salem value 1.2846165509255367367 X - needs AllOb
                4    6    7    9    11    12    14    18
Polynomial 1 - x  - x  - x  - x  - x   - x   - x   + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, 5}
Feasible primes: {5, 17}
Twists found: 0
NOT REALIZABLE

18.6 Salem value 1.2863959668362772240 X  A
                        2      3      4      5      6      7      8      9
Polynomial 1 - 2 x + 2 x  - 2 x  + 2 x  - 2 x  + 2 x  - 3 x  + 3 x  - 3 x  + 
 
        10      11      12      13      14      15      16      17    18
>    3 x   - 3 x   + 2 x   - 2 x   + 2 x   - 2 x   + 2 x   - 2 x   + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, 41}
Feasible primes: {41}
Twists found: 0
NOT REALIZABLE

20.4 Salem value 1.2920391060179294619 X A
                2    5    8    10    12    15    18    20
Polynomial 1 - x  - x  - x  + x   - x   - x   - x   + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, -3}
Feasible primes: {3}
Twists found: 0
NOT REALIZABLE

10.5 Salem value 1.2934859531254541065 ? A
                2    3    5    7    8    10
Polynomial 1 - x  - x  + x  - x  - x  + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, 1}
Feasible primes: {3, 5, 11, 17, 41, 43}
Twists found: 11
Ideal 11         Periods {5}            Min 2  OBSTRUCTED
Ideal 17         Periods {16}           Min 2  OBSTRUCTED
Ideal 5^2        Periods {6, 30}        Min 4  
Ideal 5 43       Periods {6, 22}        Min 4  
Ideal 3 5        Periods {4, 6}         Min 2  OBSTRUCTED
Ideal 3 43       Periods {4, 22}        Min 4  
Ideal 3^2        Periods {4, 12}        Min 2  OBSTRUCTED
Ideal 11 17      Periods {5, 16}        Min 4  
Ideal 3 5 11     Periods {4, 5, 6}      Min 4  
Ideal 3^2 11     Periods {4, 5, 12}     Min 4  
Ideal 3 5 17     Periods {4, 6, 16}     Min 4  

18.5 Salem value 1.2956753719440482353 X A  
                    4    5    6    8    9    10    12    13    14    17    18
Polynomial 1 - x - x  + x  - x  + x  - x  + x   - x   + x   - x   - x   + x
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, 1}
Feasible primes: {7, 11, 19}
Twists found: 0
NOT REALIZABLE
