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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 10693.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10693.e1 | 10693a3 | \([0, -1, 1, -541393, -153146123]\) | \(727057727488000/37\) | \(893090053\) | \([]\) | \(27648\) | \(1.6387\) | |
| 10693.e2 | 10693a2 | \([0, -1, 1, -6743, -204144]\) | \(1404928000/50653\) | \(1222640282557\) | \([]\) | \(9216\) | \(1.0894\) | |
| 10693.e3 | 10693a1 | \([0, -1, 1, -963, 11739]\) | \(4096000/37\) | \(893090053\) | \([]\) | \(3072\) | \(0.54008\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 10693.e have rank \(1\).
Complex multiplication
The elliptic curves in class 10693.e do not have complex multiplication.Modular form 10693.2.a.e
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
