Show commands:
SageMath
E = EllipticCurve("cl1")
E.isogeny_class()
Elliptic curves in class 69696cl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69696.gf3 | 69696cl1 | \([0, 0, 0, -4719, -122452]\) | \(140608/3\) | \(247961850048\) | \([2]\) | \(81920\) | \(0.97546\) | \(\Gamma_0(N)\)-optimal |
| 69696.gf2 | 69696cl2 | \([0, 0, 0, -10164, 212960]\) | \(21952/9\) | \(47608675209216\) | \([2, 2]\) | \(163840\) | \(1.3220\) | |
| 69696.gf4 | 69696cl3 | \([0, 0, 0, 33396, 1554608]\) | \(97336/81\) | \(-3427824615063552\) | \([2]\) | \(327680\) | \(1.6686\) | |
| 69696.gf1 | 69696cl4 | \([0, 0, 0, -140844, 20337680]\) | \(7301384/3\) | \(126956467224576\) | \([2]\) | \(327680\) | \(1.6686\) |
Rank
sage: E.rank()
The elliptic curves in class 69696cl have rank \(1\).
Complex multiplication
The elliptic curves in class 69696cl do not have complex multiplication.Modular form 69696.2.a.cl
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
