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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 42042s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 42042.s1 | 42042s1 | \([1, 1, 0, -175494, -25473888]\) | \(14812625308879/1665033084\) | \(67190090713733988\) | \([2]\) | \(501760\) | \(1.9616\) | \(\Gamma_0(N)\)-optimal |
| 42042.s2 | 42042s2 | \([1, 1, 0, 239536, -127322250]\) | \(37666121079761/195613866162\) | \(-7893725078851946334\) | \([2]\) | \(1003520\) | \(2.3082\) |
Rank
sage: E.rank()
The elliptic curves in class 42042s have rank \(1\).
Complex multiplication
The elliptic curves in class 42042s do not have complex multiplication.Modular form 42042.2.a.s
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
