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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 324900.z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
324900.z1 | 324900z1 | \([0, 0, 0, -74835300, 232392351125]\) | \(5405726654464/407253125\) | \(3491833829638225781250000\) | \([2]\) | \(49766400\) | \(3.4546\) | \(\Gamma_0(N)\)-optimal |
324900.z2 | 324900z2 | \([0, 0, 0, 71775825, 1031862815750]\) | \(298091207216/3525390625\) | \(-483633494409726562500000000\) | \([2]\) | \(99532800\) | \(3.8012\) |
Rank
sage: E.rank()
The elliptic curves in class 324900.z have rank \(0\).
Complex multiplication
The elliptic curves in class 324900.z do not have complex multiplication.Modular form 324900.2.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.