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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 31824n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31824.bi4 | 31824n1 | \([0, 0, 0, -354, 6775]\) | \(-420616192/1456611\) | \(-16989910704\) | \([2]\) | \(24576\) | \(0.64903\) | \(\Gamma_0(N)\)-optimal |
| 31824.bi3 | 31824n2 | \([0, 0, 0, -7959, 272950]\) | \(298766385232/439569\) | \(82034125056\) | \([2, 2]\) | \(49152\) | \(0.99561\) | |
| 31824.bi2 | 31824n3 | \([0, 0, 0, -10299, 99322]\) | \(161838334948/87947613\) | \(65652541314048\) | \([2]\) | \(98304\) | \(1.3422\) | |
| 31824.bi1 | 31824n4 | \([0, 0, 0, -127299, 17481778]\) | \(305612563186948/663\) | \(494926848\) | \([2]\) | \(98304\) | \(1.3422\) |
Rank
sage: E.rank()
The elliptic curves in class 31824n have rank \(1\).
Complex multiplication
The elliptic curves in class 31824n do not have complex multiplication.Modular form 31824.2.a.n
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.
