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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 111012b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
111012.a1 | 111012b1 | \([0, -1, 0, -309, 2190]\) | \(8388608/33\) | \(12877392\) | \([2]\) | \(28224\) | \(0.22165\) | \(\Gamma_0(N)\)-optimal |
111012.a2 | 111012b2 | \([0, -1, 0, -164, 4104]\) | \(-78608/1089\) | \(-6799262976\) | \([2]\) | \(56448\) | \(0.56823\) |
Rank
sage: E.rank()
The elliptic curves in class 111012b have rank \(1\).
Complex multiplication
The elliptic curves in class 111012b do not have complex multiplication.Modular form 111012.2.a.b
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.