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SageMath
E = EllipticCurve("dy1")
E.isogeny_class()
Elliptic curves in class 115920dy
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 115920.dg7 | 115920dy1 | \([0, 0, 0, -443913987, -3590451341566]\) | \(3239908336204082689644289/9880281924658790400\) | \(29502363742520353593753600\) | \([2]\) | \(31850496\) | \(3.7557\) | \(\Gamma_0(N)\)-optimal |
| 115920.dg6 | 115920dy2 | \([0, 0, 0, -632657667, -242251704574]\) | \(9378698233516887309850369/5418996968417034240000\) | \(16181038243741769568092160000\) | \([2, 2]\) | \(63700992\) | \(4.1023\) | |
| 115920.dg3 | 115920dy3 | \([0, 0, 0, -35930674947, -2621477754153214]\) | \(1718036403880129446396978632449/49057344000000\) | \(146484444266496000000\) | \([2]\) | \(95551488\) | \(4.3050\) | |
| 115920.dg8 | 115920dy4 | \([0, 0, 0, 2528430333, -1937227090174]\) | \(598672364899527954087397631/346996861747253448998400\) | \(-1036127077227510842654038425600\) | \([2]\) | \(127401984\) | \(4.4489\) | |
| 115920.dg5 | 115920dy5 | \([0, 0, 0, -6813644547, 215737500448514]\) | \(11715873038622856702991202049/46415372499833400000000\) | \(138595559638542535065600000000\) | \([2]\) | \(127401984\) | \(4.4489\) | |
| 115920.dg2 | 115920dy6 | \([0, 0, 0, -35930721027, -2621470694024446]\) | \(1718043013877225552292911401729/9180538178765625000000\) | \(27412940113183296000000000000\) | \([2, 2]\) | \(191102976\) | \(4.6516\) | |
| 115920.dg4 | 115920dy7 | \([0, 0, 0, -35300721027, -2717826548024446]\) | \(-1629247127728109256861881401729/125809119536174660320875000\) | \(-375664017989104956923567616000000\) | \([2]\) | \(382205952\) | \(4.9982\) | |
| 115920.dg1 | 115920dy8 | \([0, 0, 0, -36561458307, -2524662991783294]\) | \(1810117493172631097464564372609/125368453502655029296875000\) | \(374348196263671875000000000000000\) | \([2]\) | \(382205952\) | \(4.9982\) |
Rank
sage: E.rank()
The elliptic curves in class 115920dy have rank \(0\).
Complex multiplication
The elliptic curves in class 115920dy do not have complex multiplication.Modular form 115920.2.a.dy
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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
