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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 38640.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 38640.k1 | 38640bi8 | \([0, -1, 0, -4062384256, 93507390860800]\) | \(1810117493172631097464564372609/125368453502655029296875000\) | \(513509185546875000000000000000\) | \([2]\) | \(47775744\) | \(4.4489\) | |
| 38640.k2 | 38640bi6 | \([0, -1, 0, -3992302336, 97092837953536]\) | \(1718043013877225552292911401729/9180538178765625000000\) | \(37603484380224000000000000\) | \([2, 2]\) | \(23887872\) | \(4.1023\) | |
| 38640.k3 | 38640bi3 | \([0, -1, 0, -3992297216, 97093099438080]\) | \(1718036403880129446396978632449/49057344000000\) | \(200938881024000000\) | \([2]\) | \(11943936\) | \(3.7557\) | |
| 38640.k4 | 38640bi7 | \([0, -1, 0, -3922302336, 100661549953536]\) | \(-1629247127728109256861881401729/125809119536174660320875000\) | \(-515314153620171408674304000000\) | \([2]\) | \(47775744\) | \(4.4489\) | |
| 38640.k5 | 38640bi5 | \([0, -1, 0, -757071616, -7990025437184]\) | \(11715873038622856702991202049/46415372499833400000000\) | \(190117365759317606400000000\) | \([2]\) | \(15925248\) | \(3.8996\) | |
| 38640.k6 | 38640bi2 | \([0, -1, 0, -70295296, 8995717120]\) | \(9378698233516887309850369/5418996968417034240000\) | \(22196211582636172247040000\) | \([2, 2]\) | \(7962624\) | \(3.5530\) | |
| 38640.k7 | 38640bi1 | \([0, -1, 0, -49323776, 132996120576]\) | \(3239908336204082689644289/9880281924658790400\) | \(40469634763402405478400\) | \([2]\) | \(3981312\) | \(3.2064\) | \(\Gamma_0(N)\)-optimal |
| 38640.k8 | 38640bi4 | \([0, -1, 0, 280936704, 71655505920]\) | \(598672364899527954087397631/346996861747253448998400\) | \(-1421299145716750127097446400\) | \([2]\) | \(15925248\) | \(3.8996\) |
Rank
sage: E.rank()
The elliptic curves in class 38640.k have rank \(1\).
Complex multiplication
The elliptic curves in class 38640.k do not have complex multiplication.Modular form 38640.2.a.k
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 3 & 6 & 12 & 12 \\ 2 & 1 & 2 & 2 & 6 & 3 & 6 & 6 \\ 4 & 2 & 1 & 4 & 12 & 6 & 3 & 12 \\ 4 & 2 & 4 & 1 & 12 & 6 & 12 & 3 \\ 3 & 6 & 12 & 12 & 1 & 2 & 4 & 4 \\ 6 & 3 & 6 & 6 & 2 & 1 & 2 & 2 \\ 12 & 6 & 3 & 12 & 4 & 2 & 1 & 4 \\ 12 & 6 & 12 & 3 & 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
