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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1232.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1232.d1 | 1232f1 | \([0, -1, 0, -1429, -20323]\) | \(-78843215872/539\) | \(-2207744\) | \([]\) | \(480\) | \(0.39815\) | \(\Gamma_0(N)\)-optimal |
| 1232.d2 | 1232f2 | \([0, -1, 0, -789, -39203]\) | \(-13278380032/156590819\) | \(-641395994624\) | \([]\) | \(1440\) | \(0.94745\) | |
| 1232.d3 | 1232f3 | \([0, -1, 0, 7051, 1019197]\) | \(9463555063808/115539436859\) | \(-473249533374464\) | \([]\) | \(4320\) | \(1.4968\) |
Rank
sage: E.rank()
The elliptic curves in class 1232.d have rank \(1\).
Complex multiplication
The elliptic curves in class 1232.d do not have complex multiplication.Modular form 1232.2.a.d
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
