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SageMath
E = EllipticCurve("dn1")
E.isogeny_class()
Elliptic curves in class 76230dn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 76230.cx4 | 76230dn1 | \([1, -1, 1, 10867, 3561077]\) | \(109902239/4312000\) | \(-5568809882328000\) | \([2]\) | \(552960\) | \(1.7012\) | \(\Gamma_0(N)\)-optimal |
| 76230.cx2 | 76230dn2 | \([1, -1, 1, -294053, 58812581]\) | \(2177286259681/105875000\) | \(136734171217875000\) | \([2]\) | \(1105920\) | \(2.0478\) | |
| 76230.cx3 | 76230dn3 | \([1, -1, 1, -98033, -97454563]\) | \(-80677568161/3131816380\) | \(-4044640539559532220\) | \([2]\) | \(1658880\) | \(2.2505\) | |
| 76230.cx1 | 76230dn4 | \([1, -1, 1, -3833303, -2872013119]\) | \(4823468134087681/30382271150\) | \(39237730015697794350\) | \([2]\) | \(3317760\) | \(2.5971\) |
Rank
sage: E.rank()
The elliptic curves in class 76230dn have rank \(0\).
Complex multiplication
The elliptic curves in class 76230dn do not have complex multiplication.Modular form 76230.2.a.dn
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
