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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 382200.ek
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 382200.ek1 | 382200ek4 | \([0, -1, 0, -1055137008, -13190053025988]\) | \(69014771940559650916/9797607421875\) | \(18442859449218750000000000\) | \([2]\) | \(212336640\) | \(3.8641\) | |
| 382200.ek2 | 382200ek3 | \([0, -1, 0, -430240008, 3304154886012]\) | \(4678944235881273796/202428825314625\) | \(381048781911045066000000000\) | \([2]\) | \(212336640\) | \(3.8641\) | |
| 382200.ek3 | 382200ek2 | \([0, -1, 0, -71927508, -166459988988]\) | \(87450143958975184/25164018140625\) | \(11842086280905562500000000\) | \([2, 2]\) | \(106168320\) | \(3.5175\) | |
| 382200.ek4 | 382200ek1 | \([0, -1, 0, 11917617, -17215666488]\) | \(6364491337435136/8034291412875\) | \(-236306587608332718750000\) | \([2]\) | \(53084160\) | \(3.1710\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 382200.ek have rank \(1\).
Complex multiplication
The elliptic curves in class 382200.ek do not have complex multiplication.Modular form 382200.2.a.ek
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
