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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 230115d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
230115.d2 | 230115d1 | \([1, 1, 1, -39686, -4779142]\) | \(-46694890801/39169575\) | \(-5798502856877175\) | \([2]\) | \(1554432\) | \(1.7225\) | \(\Gamma_0(N)\)-optimal |
230115.d1 | 230115d2 | \([1, 1, 1, -730031, -240324856]\) | \(290656902035521/86293125\) | \(12774479473963125\) | \([2]\) | \(3108864\) | \(2.0690\) |
Rank
sage: E.rank()
The elliptic curves in class 230115d have rank \(0\).
Complex multiplication
The elliptic curves in class 230115d do not have complex multiplication.Modular form 230115.2.a.d
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.