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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 70602.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 70602.bf1 | 70602bn4 | \([1, 0, 0, -2259299, 1306911333]\) | \(268498407453697/252\) | \(1197026268732\) | \([2]\) | \(1105920\) | \(2.0459\) | |
| 70602.bf2 | 70602bn6 | \([1, 0, 0, -1536469, -726140497]\) | \(84448510979617/933897762\) | \(4436111719936608642\) | \([2]\) | \(2211840\) | \(2.3925\) | |
| 70602.bf3 | 70602bn3 | \([1, 0, 0, -174859, 9945869]\) | \(124475734657/63011844\) | \(299312827417630404\) | \([2, 2]\) | \(1105920\) | \(2.0459\) | |
| 70602.bf4 | 70602bn2 | \([1, 0, 0, -141239, 20401689]\) | \(65597103937/63504\) | \(301650619720464\) | \([2, 2]\) | \(552960\) | \(1.6993\) | |
| 70602.bf5 | 70602bn1 | \([1, 0, 0, -6759, 471753]\) | \(-7189057/16128\) | \(-76609681198848\) | \([2]\) | \(276480\) | \(1.3527\) | \(\Gamma_0(N)\)-optimal |
| 70602.bf6 | 70602bn5 | \([1, 0, 0, 648831, 76994235]\) | \(6359387729183/4218578658\) | \(-20038688374357888578\) | \([2]\) | \(2211840\) | \(2.3925\) |
Rank
sage: E.rank()
The elliptic curves in class 70602.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 70602.bf do not have complex multiplication.Modular form 70602.2.a.bf
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
