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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 21175.s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21175.s1 | 21175z2 | \([0, 1, 1, -215783, 38508894]\) | \(1003555225600/9317\) | \(10316021148125\) | \([]\) | \(77760\) | \(1.6606\) | |
| 21175.s2 | 21175z1 | \([0, 1, 1, -4033, -8431]\) | \(6553600/3773\) | \(4177562283125\) | \([]\) | \(25920\) | \(1.1113\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 21175.s have rank \(1\).
Complex multiplication
The elliptic curves in class 21175.s do not have complex multiplication.Modular form 21175.2.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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
