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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 12400.z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12400.z1 | 12400x2 | \([0, -1, 0, -426408, 107315312]\) | \(133974081659809/192200\) | \(12300800000000\) | \([2]\) | \(55296\) | \(1.7828\) | |
12400.z2 | 12400x1 | \([0, -1, 0, -26408, 1715312]\) | \(-31824875809/1240000\) | \(-79360000000000\) | \([2]\) | \(27648\) | \(1.4362\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12400.z have rank \(1\).
Complex multiplication
The elliptic curves in class 12400.z do not have complex multiplication.Modular form 12400.2.a.z
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.