Show commands:
SageMath
E = EllipticCurve("dc1")
E.isogeny_class()
Elliptic curves in class 44688dc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 44688.dc3 | 44688dc1 | \([0, 1, 0, -6288, 12564]\) | \(57066625/32832\) | \(15821422460928\) | \([2]\) | \(82944\) | \(1.2223\) | \(\Gamma_0(N)\)-optimal |
| 44688.dc4 | 44688dc2 | \([0, 1, 0, 25072, 125460]\) | \(3616805375/2105352\) | \(-1014548715307008\) | \([2]\) | \(165888\) | \(1.5689\) | |
| 44688.dc1 | 44688dc3 | \([0, 1, 0, -335568, -74931564]\) | \(8671983378625/82308\) | \(39663427141632\) | \([2]\) | \(248832\) | \(1.7716\) | |
| 44688.dc2 | 44688dc4 | \([0, 1, 0, -327728, -78591276]\) | \(-8078253774625/846825858\) | \(-408077170146680832\) | \([2]\) | \(497664\) | \(2.1182\) |
Rank
sage: E.rank()
The elliptic curves in class 44688dc have rank \(1\).
Complex multiplication
The elliptic curves in class 44688dc do not have complex multiplication.Modular form 44688.2.a.dc
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
