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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 323400bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
323400.bg2 | 323400bg1 | \([0, -1, 0, -139743, -20065968]\) | \(-439939433818112/143496441\) | \(-98438558526000\) | \([2]\) | \(1499136\) | \(1.6589\) | \(\Gamma_0(N)\)-optimal |
323400.bg1 | 323400bg2 | \([0, -1, 0, -2236068, -1286246268]\) | \(112650941975539952/11979\) | \(131481504000\) | \([2]\) | \(2998272\) | \(2.0055\) |
Rank
sage: E.rank()
The elliptic curves in class 323400bg have rank \(0\).
Complex multiplication
The elliptic curves in class 323400bg do not have complex multiplication.Modular form 323400.2.a.bg
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.