Show commands:
SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 31878.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31878.c1 | 31878g4 | \([1, -1, 0, -86272758, -308409484460]\) | \(97413070452067229637409633/140666577176907936\) | \(102545934761965885344\) | \([2]\) | \(2949120\) | \(3.1104\) | |
31878.c2 | 31878g3 | \([1, -1, 0, -13820598, 13345463956]\) | \(400476194988122984445793/126270124548858769248\) | \(92050920796118042781792\) | \([2]\) | \(2949120\) | \(3.1104\) | |
31878.c3 | 31878g2 | \([1, -1, 0, -5441238, -4725463820]\) | \(24439335640029940889953/902916953746891776\) | \(658226459281484104704\) | \([2, 2]\) | \(1474560\) | \(2.7639\) | |
31878.c4 | 31878g1 | \([1, -1, 0, 134442, -263804684]\) | \(368637286278891167/41443067603976192\) | \(-30211996283298643968\) | \([2]\) | \(737280\) | \(2.4173\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 31878.c have rank \(1\).
Complex multiplication
The elliptic curves in class 31878.c do not have complex multiplication.Modular form 31878.2.a.c
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.