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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 13680.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
13680.q1 | 13680l4 | \([0, 0, 0, -21963, 1252762]\) | \(784767874322/35625\) | \(53187840000\) | \([2]\) | \(24576\) | \(1.1342\) | |
13680.q2 | 13680l3 | \([0, 0, 0, -6843, -201782]\) | \(23735908082/1954815\) | \(2918523156480\) | \([2]\) | \(24576\) | \(1.1342\) | |
13680.q3 | 13680l2 | \([0, 0, 0, -1443, 17458]\) | \(445138564/81225\) | \(60634137600\) | \([2, 2]\) | \(12288\) | \(0.78768\) | |
13680.q4 | 13680l1 | \([0, 0, 0, 177, 1582]\) | \(3286064/7695\) | \(-1436071680\) | \([2]\) | \(6144\) | \(0.44110\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 13680.q have rank \(1\).
Complex multiplication
The elliptic curves in class 13680.q do not have complex multiplication.Modular form 13680.2.a.q
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.