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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 133952bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
133952.cb1 | 133952bd1 | \([0, 1, 0, -174749, -28175407]\) | \(-9221261135586623488/121324931\) | \(-7764795584\) | \([]\) | \(497664\) | \(1.4561\) | \(\Gamma_0(N)\)-optimal |
133952.cb2 | 133952bd2 | \([0, 1, 0, -164869, -31491551]\) | \(-7743965038771437568/2189290237869371\) | \(-140114575223639744\) | \([]\) | \(1492992\) | \(2.0054\) |
Rank
sage: E.rank()
The elliptic curves in class 133952bd have rank \(1\).
Complex multiplication
The elliptic curves in class 133952bd do not have complex multiplication.Modular form 133952.2.a.bd
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.