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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 135240bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
135240.z2 | 135240bd1 | \([0, -1, 0, -9649880, -11534792100]\) | \(824899990643380516/312440625\) | \(37640526940800000\) | \([2]\) | \(3317760\) | \(2.5310\) | \(\Gamma_0(N)\)-optimal |
135240.z1 | 135240bd2 | \([0, -1, 0, -9694960, -11421533108]\) | \(418257395996078018/8023271484375\) | \(1933168367340000000000\) | \([2]\) | \(6635520\) | \(2.8775\) |
Rank
sage: E.rank()
The elliptic curves in class 135240bd have rank \(0\).
Complex multiplication
The elliptic curves in class 135240bd do not have complex multiplication.Modular form 135240.2.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.