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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 117117.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 117117.p1 | 117117z1 | \([1, -1, 1, -4588557395, 119635801234850]\) | \(1382084250541230782125/19771083137421\) | \(152843917837313351288697057\) | \([2]\) | \(78653952\) | \(4.1636\) | \(\Gamma_0(N)\)-optimal |
| 117117.p2 | 117117z2 | \([1, -1, 1, -4456968080, 126819946205138]\) | \(-1266556547153680328125/165777947457789051\) | \(-1281576269967672831830612701767\) | \([2]\) | \(157307904\) | \(4.5102\) |
Rank
sage: E.rank()
The elliptic curves in class 117117.p have rank \(1\).
Complex multiplication
The elliptic curves in class 117117.p do not have complex multiplication.Modular form 117117.2.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 LMFDB 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 LMFDB labels.
