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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 3420b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3420.c2 | 3420b1 | \([0, 0, 0, -192, 781]\) | \(67108864/16245\) | \(189481680\) | \([2]\) | \(768\) | \(0.29925\) | \(\Gamma_0(N)\)-optimal |
| 3420.c1 | 3420b2 | \([0, 0, 0, -1047, -12386]\) | \(680136784/38475\) | \(7180358400\) | \([2]\) | \(1536\) | \(0.64583\) |
Rank
sage: E.rank()
The elliptic curves in class 3420b have rank \(0\).
Complex multiplication
The elliptic curves in class 3420b do not have complex multiplication.Modular form 3420.2.a.b
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
