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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 40362p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
40362.s1 | 40362p1 | \([1, 0, 1, -63927, 6678442]\) | \(-33874537/3024\) | \(-2579142497221584\) | \([3]\) | \(312480\) | \(1.7007\) | \(\Gamma_0(N)\)-optimal |
40362.s2 | 40362p2 | \([1, 0, 1, 382938, -1365128]\) | \(7281438503/4214784\) | \(-3594751498349727744\) | \([]\) | \(937440\) | \(2.2500\) |
Rank
sage: E.rank()
The elliptic curves in class 40362p have rank \(1\).
Complex multiplication
The elliptic curves in class 40362p do not have complex multiplication.Modular form 40362.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.