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SageMath
E = EllipticCurve("bh1")
E.isogeny_class()
Elliptic curves in class 44880.bh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
44880.bh1 | 44880br8 | \([0, -1, 0, -3219783720, 70322579406960]\) | \(901247067798311192691198986281/552431869440\) | \(2262760937226240\) | \([2]\) | \(15925248\) | \(3.6539\) | |
44880.bh2 | 44880br7 | \([0, -1, 0, -202588200, 1083331543152]\) | \(224494757451893010998773801/6152490825146276160000\) | \(25200602419799147151360000\) | \([2]\) | \(15925248\) | \(3.6539\) | |
44880.bh3 | 44880br6 | \([0, -1, 0, -201236520, 1098840178800]\) | \(220031146443748723000125481/172266701724057600\) | \(705604410261739929600\) | \([2, 2]\) | \(7962624\) | \(3.3074\) | |
44880.bh4 | 44880br5 | \([0, -1, 0, -39758520, 96434924400]\) | \(1696892787277117093383481/1440538624914939000\) | \(5900446207651590144000\) | \([2]\) | \(5308416\) | \(3.1046\) | |
44880.bh5 | 44880br4 | \([0, -1, 0, -26038200, -50585416848]\) | \(476646772170172569823801/5862293314453125000\) | \(24011953416000000000000\) | \([2]\) | \(5308416\) | \(3.1046\) | |
44880.bh6 | 44880br3 | \([0, -1, 0, -12492840, 17414389872]\) | \(-52643812360427830814761/1504091705903677440\) | \(-6160759627381462794240\) | \([2]\) | \(3981312\) | \(2.9608\) | |
44880.bh7 | 44880br2 | \([0, -1, 0, -3038520, 786668400]\) | \(757443433548897303481/373234243041000000\) | \(1528767459495936000000\) | \([2, 2]\) | \(2654208\) | \(2.7581\) | |
44880.bh8 | 44880br1 | \([0, -1, 0, 693960, 93920112]\) | \(9023321954633914439/6156756739584000\) | \(-25218075605336064000\) | \([2]\) | \(1327104\) | \(2.4115\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 44880.bh have rank \(1\).
Complex multiplication
The elliptic curves in class 44880.bh do not have complex multiplication.Modular form 44880.2.a.bh
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 3 & 12 & 4 & 6 & 12 \\ 4 & 1 & 2 & 12 & 3 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 3 & 12 & 6 & 1 & 4 & 12 & 2 & 4 \\ 12 & 3 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.