Show commands:
SageMath
E = EllipticCurve("ff1")
E.isogeny_class()
Elliptic curves in class 87360ff
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 87360.co4 | 87360ff1 | \([0, -1, 0, 38915, -3224483]\) | \(6364491337435136/8034291412875\) | \(-8227114406784000\) | \([2]\) | \(737280\) | \(1.7399\) | \(\Gamma_0(N)\)-optimal |
| 87360.co3 | 87360ff2 | \([0, -1, 0, -234865, -30985775]\) | \(87450143958975184/25164018140625\) | \(412287273216000000\) | \([2, 2]\) | \(1474560\) | \(2.0864\) | |
| 87360.co2 | 87360ff3 | \([0, -1, 0, -1404865, 616960225]\) | \(4678944235881273796/202428825314625\) | \(13266375495819264000\) | \([2]\) | \(2949120\) | \(2.4330\) | |
| 87360.co1 | 87360ff4 | \([0, -1, 0, -3445345, -2460034943]\) | \(69014771940559650916/9797607421875\) | \(642096000000000000\) | \([2]\) | \(2949120\) | \(2.4330\) |
Rank
sage: E.rank()
The elliptic curves in class 87360ff have rank \(0\).
Complex multiplication
The elliptic curves in class 87360ff do not have complex multiplication.Modular form 87360.2.a.ff
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.
