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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 693c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 693.b1 | 693c1 | \([0, 0, 1, -804, -8775]\) | \(-78843215872/539\) | \(-392931\) | \([]\) | \(200\) | \(0.25431\) | \(\Gamma_0(N)\)-optimal |
| 693.b2 | 693c2 | \([0, 0, 1, -444, -16650]\) | \(-13278380032/156590819\) | \(-114154707051\) | \([3]\) | \(600\) | \(0.80361\) | |
| 693.b3 | 693c3 | \([0, 0, 1, 3966, 430965]\) | \(9463555063808/115539436859\) | \(-84228249470211\) | \([3]\) | \(1800\) | \(1.3529\) |
Rank
sage: E.rank()
The elliptic curves in class 693c have rank \(0\).
Complex multiplication
The elliptic curves in class 693c do not have complex multiplication.Modular form 693.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
