Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1008.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1008.e1 | 1008g3 | \([0, 0, 0, -1371, 19514]\) | \(381775972/567\) | \(423263232\) | \([2]\) | \(512\) | \(0.55626\) | |
1008.e2 | 1008g2 | \([0, 0, 0, -111, 110]\) | \(810448/441\) | \(82301184\) | \([2, 2]\) | \(256\) | \(0.20969\) | |
1008.e3 | 1008g1 | \([0, 0, 0, -66, -205]\) | \(2725888/21\) | \(244944\) | \([2]\) | \(128\) | \(-0.13688\) | \(\Gamma_0(N)\)-optimal |
1008.e4 | 1008g4 | \([0, 0, 0, 429, 866]\) | \(11696828/7203\) | \(-5377010688\) | \([4]\) | \(512\) | \(0.55626\) |
Rank
sage: E.rank()
The elliptic curves in class 1008.e have rank \(1\).
Complex multiplication
The elliptic curves in class 1008.e do not have complex multiplication.Modular form 1008.2.a.e
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.