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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 100017.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 100017.a1 | 100017a1 | \([1, -1, 1, -1841, -27840]\) | \(946098541513/72912393\) | \(53153134497\) | \([2]\) | \(74752\) | \(0.80256\) | \(\Gamma_0(N)\)-optimal |
| 100017.a2 | 100017a2 | \([1, -1, 1, 1804, -126984]\) | \(891110287367/10003400289\) | \(-7292478810681\) | \([2]\) | \(149504\) | \(1.1491\) |
Rank
sage: E.rank()
The elliptic curves in class 100017.a have rank \(1\).
Complex multiplication
The elliptic curves in class 100017.a do not have complex multiplication.Modular form 100017.2.a.a
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.
