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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 457776.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
457776.k1 | 457776k2 | \([0, 0, 0, -9121707, 6678977850]\) | \(86265529686/30116537\) | \(29303520912956536215552\) | \([2]\) | \(58392576\) | \(3.0125\) | \(\Gamma_0(N)\)-optimal* |
457776.k2 | 457776k1 | \([0, 0, 0, -3815667, -2792303550]\) | \(12628458252/384659\) | \(187137436333681861632\) | \([2]\) | \(29196288\) | \(2.6659\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 457776.k have rank \(0\).
Complex multiplication
The elliptic curves in class 457776.k do not have complex multiplication.Modular form 457776.2.a.k
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.