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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 182070ef
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 182070.n2 | 182070ef1 | \([1, -1, 0, -14543775, 21351859661]\) | \(2564747659190919557979/14336000000000\) | \(1901684736000000000\) | \([2]\) | \(7925760\) | \(2.6997\) | \(\Gamma_0(N)\)-optimal |
| 182070.n1 | 182070ef2 | \([1, -1, 0, -14804895, 20545573325]\) | \(2705385731431382528859/191406250000000000\) | \(25390230468750000000000\) | \([2]\) | \(15851520\) | \(3.0463\) |
Rank
sage: E.rank()
The elliptic curves in class 182070ef have rank \(1\).
Complex multiplication
The elliptic curves in class 182070ef do not have complex multiplication.Modular form 182070.2.a.ef
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
