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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 175c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 175.c2 | 175c1 | \([0, 1, 1, 42, -131]\) | \(4096/7\) | \(-13671875\) | \([]\) | \(40\) | \(0.054193\) | \(\Gamma_0(N)\)-optimal |
| 175.c1 | 175c2 | \([0, 1, 1, -3708, 86119]\) | \(-2887553024/16807\) | \(-32826171875\) | \([]\) | \(200\) | \(0.85891\) |
Rank
sage: E.rank()
The elliptic curves in class 175c have rank \(0\).
Complex multiplication
The elliptic curves in class 175c do not have complex multiplication.Modular form 175.2.a.c
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
