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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 68970w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68970.u2 | 68970w1 | \([1, 0, 1, -113864, -103033438]\) | \(-92155535561809/2534906137500\) | \(-4490740851855637500\) | \([2]\) | \(1382400\) | \(2.2596\) | \(\Gamma_0(N)\)-optimal |
68970.u1 | 68970w2 | \([1, 0, 1, -4066934, -3142153654]\) | \(4199221866816810289/23034902343750\) | \(40807734630996093750\) | \([2]\) | \(2764800\) | \(2.6061\) |
Rank
sage: E.rank()
The elliptic curves in class 68970w have rank \(1\).
Complex multiplication
The elliptic curves in class 68970w do not have complex multiplication.Modular form 68970.2.a.w
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.