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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 88445p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
88445.k3 | 88445p1 | \([1, -1, 1, -250963, -48239838]\) | \(315821241/665\) | \(3680709067756385\) | \([2]\) | \(552960\) | \(1.8719\) | \(\Gamma_0(N)\)-optimal |
88445.k2 | 88445p2 | \([1, -1, 1, -339408, -11163694]\) | \(781229961/442225\) | \(2447671530057996025\) | \([2, 2]\) | \(1105920\) | \(2.2184\) | |
88445.k4 | 88445p3 | \([1, -1, 1, 1341047, -89808988]\) | \(48188806119/28511875\) | \(-157810401280055006875\) | \([2]\) | \(2211840\) | \(2.5650\) | |
88445.k1 | 88445p4 | \([1, -1, 1, -3434983, 2439293476]\) | \(809818183161/4561235\) | \(25245983495741044715\) | \([2]\) | \(2211840\) | \(2.5650\) |
Rank
sage: E.rank()
The elliptic curves in class 88445p have rank \(1\).
Complex multiplication
The elliptic curves in class 88445p do not have complex multiplication.Modular form 88445.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.