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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 115920.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
115920.p1 | 115920s4 | \([0, 0, 0, -325083, -66919718]\) | \(5089545532199524/353759765625\) | \(264080250000000000\) | \([2]\) | \(1179648\) | \(2.0908\) | |
115920.p2 | 115920s2 | \([0, 0, 0, -64263, 5014438]\) | \(157267580823376/32806265625\) | \(6122436516000000\) | \([2, 2]\) | \(589824\) | \(1.7442\) | |
115920.p3 | 115920s1 | \([0, 0, 0, -60618, 5744167]\) | \(2111937254864896/132040125\) | \(1540116018000\) | \([2]\) | \(294912\) | \(1.3976\) | \(\Gamma_0(N)\)-optimal |
115920.p4 | 115920s3 | \([0, 0, 0, 138237, 30245938]\) | \(391353415004156/755885521125\) | \(-564265517977728000\) | \([2]\) | \(1179648\) | \(2.0908\) |
Rank
sage: E.rank()
The elliptic curves in class 115920.p have rank \(1\).
Complex multiplication
The elliptic curves in class 115920.p do not have complex multiplication.Modular form 115920.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.