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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 21675j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21675.j2 | 21675j1 | \([0, -1, 1, -110783, 903593]\) | \(115220905984/66430125\) | \(86692351095703125\) | \([]\) | \(165888\) | \(1.9396\) | \(\Gamma_0(N)\)-optimal |
| 21675.j1 | 21675j2 | \([0, -1, 1, -5963033, -5602625782]\) | \(17968412610002944/158203125\) | \(206457550048828125\) | \([]\) | \(497664\) | \(2.4890\) |
Rank
sage: E.rank()
The elliptic curves in class 21675j have rank \(0\).
Complex multiplication
The elliptic curves in class 21675j do not have complex multiplication.Modular form 21675.2.a.j
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
