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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 11858z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
11858.y2 | 11858z1 | \([1, 0, 0, -5783, -169751]\) | \(73622481625/512\) | \(148746752\) | \([]\) | \(10368\) | \(0.74803\) | \(\Gamma_0(N)\)-optimal |
11858.y1 | 11858z2 | \([1, 0, 0, -8478, 3268]\) | \(231968823625/134217728\) | \(38993068556288\) | \([]\) | \(31104\) | \(1.2973\) |
Rank
sage: E.rank()
The elliptic curves in class 11858z have rank \(1\).
Complex multiplication
The elliptic curves in class 11858z do not have complex multiplication.Modular form 11858.2.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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.