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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 8624.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8624.j1 | 8624r3 | \([0, -1, 0, -6131141, -5841282131]\) | \(-52893159101157376/11\) | \(-5300793344\) | \([]\) | \(72000\) | \(2.1628\) | |
| 8624.j2 | 8624r2 | \([0, -1, 0, -8101, -505651]\) | \(-122023936/161051\) | \(-77608915349504\) | \([]\) | \(14400\) | \(1.3581\) | |
| 8624.j3 | 8624r1 | \([0, -1, 0, -261, 3949]\) | \(-4096/11\) | \(-5300793344\) | \([]\) | \(2880\) | \(0.55337\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8624.j have rank \(1\).
Complex multiplication
The elliptic curves in class 8624.j do not have complex multiplication.Modular form 8624.2.a.j
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.
\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
