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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 7854a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7854.a4 | 7854a1 | \([1, 1, 0, -366, 180]\) | \(5445273626857/3103398144\) | \(3103398144\) | \([2]\) | \(4608\) | \(0.51113\) | \(\Gamma_0(N)\)-optimal |
| 7854.a2 | 7854a2 | \([1, 1, 0, -4286, 106020]\) | \(8710408612492777/19986042384\) | \(19986042384\) | \([2, 2]\) | \(9216\) | \(0.85770\) | |
| 7854.a1 | 7854a3 | \([1, 1, 0, -68546, 6879024]\) | \(35618855581745079337/188166132\) | \(188166132\) | \([2]\) | \(18432\) | \(1.2043\) | |
| 7854.a3 | 7854a4 | \([1, 1, 0, -2746, 185176]\) | \(-2291249615386537/13671036998388\) | \(-13671036998388\) | \([2]\) | \(18432\) | \(1.2043\) |
Rank
sage: E.rank()
The elliptic curves in class 7854a have rank \(1\).
Complex multiplication
The elliptic curves in class 7854a do not have complex multiplication.Modular form 7854.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
