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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 485100.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485100.d1 | 485100d2 | \([0, 0, 0, -5692575, 5218487750]\) | \(59466754384/121275\) | \(41605145297100000000\) | \([2]\) | \(17694720\) | \(2.6511\) | \(\Gamma_0(N)\)-optimal* |
485100.d2 | 485100d1 | \([0, 0, 0, -235200, 137671625]\) | \(-67108864/343035\) | \(-7355195329308750000\) | \([2]\) | \(8847360\) | \(2.3045\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 485100.d have rank \(2\).
Complex multiplication
The elliptic curves in class 485100.d do not have complex multiplication.Modular form 485100.2.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 LMFDB 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 LMFDB labels.