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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 412224bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
412224.bd2 | 412224bd1 | \([0, -1, 0, -377, -903]\) | \(1450571968/727833\) | \(2981203968\) | \([2]\) | \(221184\) | \(0.51070\) | \(\Gamma_0(N)\)-optimal |
412224.bd1 | 412224bd2 | \([0, -1, 0, -4897, -130175]\) | \(396417457736/367137\) | \(12030345216\) | \([2]\) | \(442368\) | \(0.85728\) |
Rank
sage: E.rank()
The elliptic curves in class 412224bd have rank \(0\).
Complex multiplication
The elliptic curves in class 412224bd do not have complex multiplication.Modular form 412224.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.