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SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 8400bl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8400.k5 | 8400bl1 | \([0, -1, 0, -1608, -54288]\) | \(-7189057/16128\) | \(-1032192000000\) | \([2]\) | \(12288\) | \(0.99382\) | \(\Gamma_0(N)\)-optimal |
| 8400.k4 | 8400bl2 | \([0, -1, 0, -33608, -2358288]\) | \(65597103937/63504\) | \(4064256000000\) | \([2, 2]\) | \(24576\) | \(1.3404\) | |
| 8400.k1 | 8400bl3 | \([0, -1, 0, -537608, -151542288]\) | \(268498407453697/252\) | \(16128000000\) | \([2]\) | \(49152\) | \(1.6870\) | |
| 8400.k3 | 8400bl4 | \([0, -1, 0, -41608, -1142288]\) | \(124475734657/63011844\) | \(4032758016000000\) | \([2, 2]\) | \(49152\) | \(1.6870\) | |
| 8400.k2 | 8400bl5 | \([0, -1, 0, -365608, 84393712]\) | \(84448510979617/933897762\) | \(59769456768000000\) | \([2]\) | \(98304\) | \(2.0335\) | |
| 8400.k6 | 8400bl6 | \([0, -1, 0, 154392, -8982288]\) | \(6359387729183/4218578658\) | \(-269989034112000000\) | \([2]\) | \(98304\) | \(2.0335\) |
Rank
sage: E.rank()
The elliptic curves in class 8400bl have rank \(0\).
Complex multiplication
The elliptic curves in class 8400bl do not have complex multiplication.Modular form 8400.2.a.bl
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.
