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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 5200ba
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5200.bd1 | 5200ba1 | \([0, -1, 0, -336408, 75203312]\) | \(65787589563409/10400000\) | \(665600000000000\) | \([2]\) | \(46080\) | \(1.8548\) | \(\Gamma_0(N)\)-optimal |
5200.bd2 | 5200ba2 | \([0, -1, 0, -304408, 90051312]\) | \(-48743122863889/26406250000\) | \(-1690000000000000000\) | \([2]\) | \(92160\) | \(2.2014\) |
Rank
sage: E.rank()
The elliptic curves in class 5200ba have rank \(1\).
Complex multiplication
The elliptic curves in class 5200ba do not have complex multiplication.Modular form 5200.2.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 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.