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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 10008.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10008.f1 | 10008f1 | \([0, 0, 0, -11235, -458354]\) | \(210094874500/3753\) | \(2801599488\) | \([2]\) | \(10752\) | \(0.93982\) | \(\Gamma_0(N)\)-optimal |
| 10008.f2 | 10008f2 | \([0, 0, 0, -10875, -489098]\) | \(-95269531250/14085009\) | \(-21028805756928\) | \([2]\) | \(21504\) | \(1.2864\) |
Rank
sage: E.rank()
The elliptic curves in class 10008.f have rank \(1\).
Complex multiplication
The elliptic curves in class 10008.f do not have complex multiplication.Modular form 10008.2.a.f
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.
