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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 465690.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
465690.p1 | 465690p1 | \([1, 0, 1, -9615604, 11512499906]\) | \(-5789279907940249/21466890000\) | \(-364584279609212490000\) | \([3]\) | \(45702144\) | \(2.8056\) | \(\Gamma_0(N)\)-optimal |
465690.p2 | 465690p2 | \([1, 0, 1, 21352781, 60244350542]\) | \(63395476613331191/129000000000000\) | \(-2190879632289000000000000\) | \([]\) | \(137106432\) | \(3.3549\) |
Rank
sage: E.rank()
The elliptic curves in class 465690.p have rank \(1\).
Complex multiplication
The elliptic curves in class 465690.p do not have complex multiplication.Modular form 465690.2.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 LMFDB 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 LMFDB labels.