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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 3675p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3675.l1 | 3675p1 | \([1, 0, 1, -22076, -1223827]\) | \(5177717/189\) | \(43429025390625\) | \([2]\) | \(11520\) | \(1.3865\) | \(\Gamma_0(N)\)-optimal |
3675.l2 | 3675p2 | \([1, 0, 1, 8549, -4347577]\) | \(300763/35721\) | \(-8208085798828125\) | \([2]\) | \(23040\) | \(1.7330\) |
Rank
sage: E.rank()
The elliptic curves in class 3675p have rank \(0\).
Complex multiplication
The elliptic curves in class 3675p do not have complex multiplication.Modular form 3675.2.a.p
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.