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SageMath
E = EllipticCurve("gb1")
E.isogeny_class()
Elliptic curves in class 121968.gb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121968.gb1 | 121968gd2 | \([0, 0, 0, -48576, 4120688]\) | \(35084566528/1029\) | \(371781881856\) | \([]\) | \(331776\) | \(1.3199\) | |
| 121968.gb2 | 121968gd1 | \([0, 0, 0, -1056, -4048]\) | \(360448/189\) | \(68286468096\) | \([]\) | \(110592\) | \(0.77063\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121968.gb have rank \(1\).
Complex multiplication
The elliptic curves in class 121968.gb do not have complex multiplication.Modular form 121968.2.a.gb
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
