Show commands:
SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 29400.ce
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
29400.ce1 | 29400n6 | \([0, -1, 0, -470808, 124497612]\) | \(3065617154/9\) | \(33882912000000\) | \([2]\) | \(196608\) | \(1.8255\) | |
29400.ce2 | 29400n4 | \([0, -1, 0, -78808, -8488388]\) | \(28756228/3\) | \(5647152000000\) | \([2]\) | \(98304\) | \(1.4789\) | |
29400.ce3 | 29400n3 | \([0, -1, 0, -29808, 1899612]\) | \(1556068/81\) | \(152473104000000\) | \([2, 2]\) | \(98304\) | \(1.4789\) | |
29400.ce4 | 29400n2 | \([0, -1, 0, -5308, -109388]\) | \(35152/9\) | \(4235364000000\) | \([2, 2]\) | \(49152\) | \(1.1323\) | |
29400.ce5 | 29400n1 | \([0, -1, 0, 817, -11388]\) | \(2048/3\) | \(-88236750000\) | \([2]\) | \(24576\) | \(0.78575\) | \(\Gamma_0(N)\)-optimal |
29400.ce6 | 29400n5 | \([0, -1, 0, 19192, 7485612]\) | \(207646/6561\) | \(-24700642848000000\) | \([2]\) | \(196608\) | \(1.8255\) |
Rank
sage: E.rank()
The elliptic curves in class 29400.ce have rank \(0\).
Complex multiplication
The elliptic curves in class 29400.ce do not have complex multiplication.Modular form 29400.2.a.ce
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.