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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 450800bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
450800.bd1 | 450800bd1 | \([0, 1, 0, -703908, 227024188]\) | \(28113694476208/7604375\) | \(10433202500000000\) | \([2]\) | \(4128768\) | \(2.0568\) | \(\Gamma_0(N)\)-optimal |
450800.bd2 | 450800bd2 | \([0, 1, 0, -616408, 285649188]\) | \(-4719707817052/3700897225\) | \(-20310523970800000000\) | \([2]\) | \(8257536\) | \(2.4034\) |
Rank
sage: E.rank()
The elliptic curves in class 450800bd have rank \(0\).
Complex multiplication
The elliptic curves in class 450800bd do not have complex multiplication.Modular form 450800.2.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.