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21 |
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File format:
Data fields (and metafields combining several fields):
N: Conductor
CLASSCODE: Letter code of isogeny class (a, b, ..., z, ba, bb, ...)
in the Cremona labelling scheme
NCURVE: Number of the curve in its isogeny class (1,2,...)
CREMONA_LABEL: Concatenation of (N,CLASSCODE,NCURVE) with no
whitespace separation
TOR_GROWTH: Sequence of 0 or more TOR_NF_d separated by whitespace
TOR_NF_d: [n][a0,…,ad-1,1] or [n1,n2][a0,…,ad-1,1] where [a0,…,ad-1,1] integer coeficients of the defining polynomial
of a number field K of degree d where the torsion grows to [n] or [n1,n2] with n, n1, n2 integers>1 and n1 dividing n2.
TOR_NF_SMALL_dd: Concatenation of "dd:" with 1 or more [n] or [n1,n2] (with n, n1, n2 integers>1 and n1 dividing n2) where the torsion grows to [n] or [n1,n2] over a number field of degree dd
TOR_GROWTH_SMALL_d: Sequence of 0 or more TOR_NF_SMALL_dd for each divisor dd of a given in separated by whitespace
For each d in {2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21} two files.
Each file consists of lines, one per curve, with certain data fields
on each line, separated by whitespace:
File growthdsage.000000-399999.txt (Only for elliptic curves with primitive torsion growth over number fields of degree d)
A. growth_torsion: CREMONA_LABEL TOR TOR_GROWTH
File onlygrowthd.000000-399999.txt
B. onlygrowth_torsion: CREMONA_LABEL TOR TOR_GROWTH_SMALL
Example: d=4
*** 3rd line of the file onlygrowth4.000000-399999.txt
14a1 [6] 2:[2,6][3,6] 4:[12][6,6]
Then for the elliptic curve 14a1 we have the following info about primitive torsion growth over number fields of degree dividing d=4:
* over Q: E(Q)_tors = Z/6Z
* over quadratic number fields: there exits 2 quadratic fields K1 and K2 such that E(K1)_tors = Z/2Z+Z/6Z and E(K2)_tors = Z/3Z+Z/6Z.
* over quartic fields: there exits 2 quartic fields L1 and L2 such that E(L1)_tors = Z/12Z and E(L2)_tors = Z/6Z+Z/6Z.
*** 3rd line of the file growth4sage.000000-399999.txt
14a1 [12][-1, 2, -1, -2, 1] [6,6][4, -2, -1, -1, 1]
Then for the elliptic curve 14a1 we have the following info about primitive torsion growth over number fields of degree d=4:
* Over the number field of degree 4 with the coefficients of defining polynomial equal to
** [-1, 2, -1, -2, 1] it has primitive torsion growth Z/12Z
** [4, -2, -1, -1, 1] it has primitive torsion growth Z/6Z+Z/6Z
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Some references:
* algorithm ** An algorithm for determining torsion growth of elliptic curves (joint with F. Najman). arXiv: 1904.07071
* d ** Growth of torsion groups of elliptic curves upon base changes (joint with F. Najman). arXiv: 1609.02515
* d=6 ** On the torsion of rational elliptic curves over sextic fields (joint with H.B. Daniels). Mathematics of Computation, to appear.
* d=4 ** On the torsion of rational elliptic curves over quartic fields (joint with Á. Lozano-Robledo). Mathematics of Computation 87 (2018) 1457-1478.
* d=5 ** Complete classification of the torsion structures of rational elliptic curves over quintic number fields. Journal of Algebra 478 (2017) 484–505
* d=3 ** Torsion of rational elliptic curves over cubic fields (joint with F. Najman and J.M. Tornero). Rocky Mountain Journal of Mathematics 46, 1899-1917 (2016).
* d=2 ** Torsion of rational elliptic curves over quadratic fields II (joint with J.M. Tornero). Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemática, 110 (1) (2016), 121-143.
* d=2 ** Torsion of rational elliptic curves over quadratic fields (joint with J.M. Tornero). Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas. 108 (2) (2014), 923-934.
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d<24 or (d,2*3*5*7)=1
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More links:
http://www.lmfdb.org/EllipticCurve/Q/
http://johncremona.github.io/ecdata/