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SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 70560bw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
70560.ci3 | 70560bw1 | \([0, 0, 0, -31017, 2068976]\) | \(601211584/11025\) | \(60516574977600\) | \([2, 2]\) | \(294912\) | \(1.4393\) | \(\Gamma_0(N)\)-optimal |
70560.ci4 | 70560bw2 | \([0, 0, 0, -147, 6001814]\) | \(-8/354375\) | \(-15561404994240000\) | \([2]\) | \(589824\) | \(1.7858\) | |
70560.ci2 | 70560bw3 | \([0, 0, 0, -64092, -3117184]\) | \(82881856/36015\) | \(12651998608650240\) | \([2]\) | \(589824\) | \(1.7858\) | |
70560.ci1 | 70560bw4 | \([0, 0, 0, -494067, 133667786]\) | \(303735479048/105\) | \(4610786664960\) | \([2]\) | \(589824\) | \(1.7858\) |
Rank
sage: E.rank()
The elliptic curves in class 70560bw have rank \(2\).
Complex multiplication
The elliptic curves in class 70560bw do not have complex multiplication.Modular form 70560.2.a.bw
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.