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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 6864.m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6864.m1 | 6864d3 | \([0, -1, 0, -69752, 7113888]\) | \(36652193922790372/93308787\) | \(95548197888\) | \([4]\) | \(21504\) | \(1.3445\) | |
| 6864.m2 | 6864d2 | \([0, -1, 0, -4412, 109440]\) | \(37109806448848/1803785841\) | \(461769175296\) | \([2, 2]\) | \(10752\) | \(0.99791\) | |
| 6864.m3 | 6864d1 | \([0, -1, 0, -767, -5742]\) | \(3122884507648/835956693\) | \(13375307088\) | \([2]\) | \(5376\) | \(0.65133\) | \(\Gamma_0(N)\)-optimal |
| 6864.m4 | 6864d4 | \([0, -1, 0, 2608, 418320]\) | \(1915049403068/75239967231\) | \(-77045726444544\) | \([2]\) | \(21504\) | \(1.3445\) |
Rank
sage: E.rank()
The elliptic curves in class 6864.m have rank \(0\).
Complex multiplication
The elliptic curves in class 6864.m do not have complex multiplication.Modular form 6864.2.a.m
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.
