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SageMath
E = EllipticCurve("dj1")
E.isogeny_class()
Elliptic curves in class 243936dj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 243936.dj3 | 243936dj1 | \([0, 0, 0, -15609, -692120]\) | \(5088448/441\) | \(36450391957056\) | \([2, 2]\) | \(655360\) | \(1.3430\) | \(\Gamma_0(N)\)-optimal |
| 243936.dj2 | 243936dj2 | \([0, 0, 0, -53724, 4003648]\) | \(3241792/567\) | \(2999346538180608\) | \([2]\) | \(1310720\) | \(1.6896\) | |
| 243936.dj4 | 243936dj3 | \([0, 0, 0, 17061, -3207710]\) | \(830584/7203\) | \(-4762851215721984\) | \([2]\) | \(1310720\) | \(1.6896\) | |
| 243936.dj1 | 243936dj4 | \([0, 0, 0, -244299, -46475858]\) | \(2438569736/21\) | \(13885863602688\) | \([2]\) | \(1310720\) | \(1.6896\) |
Rank
sage: E.rank()
The elliptic curves in class 243936dj have rank \(1\).
Complex multiplication
The elliptic curves in class 243936dj do not have complex multiplication.Modular form 243936.2.a.dj
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
