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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 13860.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
13860.x1 | 13860x4 | \([0, 0, 0, -446367, -60520826]\) | \(52702650535889104/22020583921875\) | \(4109569453836000000\) | \([6]\) | \(248832\) | \(2.2687\) | |
13860.x2 | 13860x2 | \([0, 0, 0, -384807, -91878194]\) | \(33766427105425744/9823275\) | \(1833258873600\) | \([2]\) | \(82944\) | \(1.7194\) | |
13860.x3 | 13860x1 | \([0, 0, 0, -23952, -1447931]\) | \(-130287139815424/2250652635\) | \(-26251612334640\) | \([2]\) | \(41472\) | \(1.3728\) | \(\Gamma_0(N)\)-optimal |
13860.x4 | 13860x3 | \([0, 0, 0, 92688, -6938759]\) | \(7549996227362816/6152409907875\) | \(-71761709165454000\) | \([6]\) | \(124416\) | \(1.9222\) |
Rank
sage: E.rank()
The elliptic curves in class 13860.x have rank \(0\).
Complex multiplication
The elliptic curves in class 13860.x do not have complex multiplication.Modular form 13860.2.a.x
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.