Show commands:
SageMath
E = EllipticCurve("cl1")
E.isogeny_class()
Elliptic curves in class 34320cl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34320.cm4 | 34320cl1 | \([0, 1, 0, 1040, 17108]\) | \(30342134159/47190000\) | \(-193290240000\) | \([2]\) | \(49152\) | \(0.85089\) | \(\Gamma_0(N)\)-optimal |
| 34320.cm3 | 34320cl2 | \([0, 1, 0, -6960, 167508]\) | \(9104453457841/2226896100\) | \(9121366425600\) | \([2, 2]\) | \(98304\) | \(1.1975\) | |
| 34320.cm2 | 34320cl3 | \([0, 1, 0, -38160, -2740332]\) | \(1500376464746641/83599963590\) | \(342425450864640\) | \([2]\) | \(196608\) | \(1.5440\) | |
| 34320.cm1 | 34320cl4 | \([0, 1, 0, -103760, 12828948]\) | \(30161840495801041/2799263610\) | \(11465783746560\) | \([4]\) | \(196608\) | \(1.5440\) |
Rank
sage: E.rank()
The elliptic curves in class 34320cl have rank \(0\).
Complex multiplication
The elliptic curves in class 34320cl do not have complex multiplication.Modular form 34320.2.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.
