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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 123200.ba
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
123200.ba1 | 123200ew2 | \([0, 1, 0, -82433, -8436737]\) | \(15124197817/1294139\) | \(5300793344000000\) | \([2]\) | \(786432\) | \(1.7584\) | |
123200.ba2 | 123200ew1 | \([0, 1, 0, 5567, -604737]\) | \(4657463/41503\) | \(-169996288000000\) | \([2]\) | \(393216\) | \(1.4118\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 123200.ba have rank \(1\).
Complex multiplication
The elliptic curves in class 123200.ba do not have complex multiplication.Modular form 123200.2.a.ba
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.