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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 57960p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 57960.s3 | 57960p1 | \([0, 0, 0, -60618, -5744167]\) | \(2111937254864896/132040125\) | \(1540116018000\) | \([2]\) | \(147456\) | \(1.3976\) | \(\Gamma_0(N)\)-optimal |
| 57960.s2 | 57960p2 | \([0, 0, 0, -64263, -5014438]\) | \(157267580823376/32806265625\) | \(6122436516000000\) | \([2, 2]\) | \(294912\) | \(1.7442\) | |
| 57960.s4 | 57960p3 | \([0, 0, 0, 138237, -30245938]\) | \(391353415004156/755885521125\) | \(-564265517977728000\) | \([2]\) | \(589824\) | \(2.0908\) | |
| 57960.s1 | 57960p4 | \([0, 0, 0, -325083, 66919718]\) | \(5089545532199524/353759765625\) | \(264080250000000000\) | \([2]\) | \(589824\) | \(2.0908\) |
Rank
sage: E.rank()
The elliptic curves in class 57960p have rank \(1\).
Complex multiplication
The elliptic curves in class 57960p do not have complex multiplication.Modular form 57960.2.a.p
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
