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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 364650t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 364650.t4 | 364650t1 | \([1, 1, 0, -3308535750, 73245928756500]\) | \(256336931635587146090976858721/6966078430857972940800\) | \(108844975482155827200000000\) | \([2]\) | \(259522560\) | \(4.1000\) | \(\Gamma_0(N)\)-optimal |
| 364650.t3 | 364650t2 | \([1, 1, 0, -3439607750, 67127881012500]\) | \(288025170744824162562114015841/42089811414708944240640000\) | \(657653303354827253760000000000\) | \([2, 2]\) | \(519045120\) | \(4.4465\) | |
| 364650.t5 | 364650t3 | \([1, 1, 0, 5753992250, 364458098612500]\) | \(1348377521288200270907278880159/4452738450438853168316236800\) | \(-69574038288107080754941200000000\) | \([2]\) | \(1038090240\) | \(4.7931\) | |
| 364650.t2 | 364650t4 | \([1, 1, 0, -14730359750, -621754770763500]\) | \(22622664067287084590070422475361/2403894445280029587600000000\) | \(37560850707500462306250000000000\) | \([2, 2]\) | \(1038090240\) | \(4.7931\) | |
| 364650.t6 | 364650t5 | \([1, 1, 0, 19200108250, -3066207476887500]\) | \(50097345876551739554894890530719/288929095004409653320312500000\) | \(-4514517109443900833129882812500000\) | \([2]\) | \(2076180480\) | \(5.1397\) | |
| 364650.t1 | 364650t6 | \([1, 1, 0, -229312859750, -42265565963263500]\) | \(85347129192082387928749867185675361/1142294076320397505851060000\) | \(17848344942506211028922812500000\) | \([2]\) | \(2076180480\) | \(5.1397\) |
Rank
sage: E.rank()
The elliptic curves in class 364650t have rank \(0\).
Complex multiplication
The elliptic curves in class 364650t do not have complex multiplication.Modular form 364650.2.a.t
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
