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SageMath
E = EllipticCurve("dt1")
E.isogeny_class()
Elliptic curves in class 479808.dt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
479808.dt1 | 479808dt4 | \([0, 0, 0, -2244396, -1294163696]\) | \(444893916104/9639\) | \(27089293813972992\) | \([2]\) | \(5505024\) | \(2.2696\) | |
479808.dt2 | 479808dt2 | \([0, 0, 0, -145236, -18714080]\) | \(964430272/127449\) | \(44772582831427584\) | \([2, 2]\) | \(2752512\) | \(1.9230\) | |
479808.dt3 | 479808dt1 | \([0, 0, 0, -37191, 2462740]\) | \(1036433728/122451\) | \(672137426084544\) | \([2]\) | \(1376256\) | \(1.5764\) | \(\Gamma_0(N)\)-optimal* |
479808.dt4 | 479808dt3 | \([0, 0, 0, 225204, -98580944]\) | \(449455096/1753941\) | \(-4929248166964789248\) | \([2]\) | \(5505024\) | \(2.2696\) |
Rank
sage: E.rank()
The elliptic curves in class 479808.dt have rank \(1\).
Complex multiplication
The elliptic curves in class 479808.dt do not have complex multiplication.Modular form 479808.2.a.dt
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.