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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1122.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1122.a1 | 1122c3 | \([1, 1, 0, -1586306, 768343356]\) | \(441453577446719855661097/4354701912\) | \(4354701912\) | \([2]\) | \(10752\) | \(1.8813\) | |
1122.a2 | 1122c2 | \([1, 1, 0, -99146, 11973780]\) | \(107784459654566688937/10704361149504\) | \(10704361149504\) | \([2, 2]\) | \(5376\) | \(1.5348\) | |
1122.a3 | 1122c4 | \([1, 1, 0, -91666, 13866220]\) | \(-85183593440646799657/34223681512621656\) | \(-34223681512621656\) | \([2]\) | \(10752\) | \(1.8813\) | |
1122.a4 | 1122c1 | \([1, 1, 0, -6666, 154836]\) | \(32765849647039657/8229948198912\) | \(8229948198912\) | \([2]\) | \(2688\) | \(1.1882\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1122.a have rank \(0\).
Complex multiplication
The elliptic curves in class 1122.a do not have complex multiplication.Modular form 1122.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.