Show commands:
SageMath
E = EllipticCurve("dx1")
E.isogeny_class()
Elliptic curves in class 397800.dx
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
397800.dx1 | 397800dx3 | \([0, 0, 0, -20304075, 26251829750]\) | \(79364416584061444/20404090514925\) | \(237993311766085200000000\) | \([2]\) | \(37748736\) | \(3.1949\) | |
397800.dx2 | 397800dx2 | \([0, 0, 0, -7141575, -7009807750]\) | \(13813960087661776/714574355625\) | \(2083698821002500000000\) | \([2, 2]\) | \(18874368\) | \(2.8483\) | |
397800.dx3 | 397800dx1 | \([0, 0, 0, -7050450, -7205635375]\) | \(212670222886967296/616241925\) | \(112310090831250000\) | \([2]\) | \(9437184\) | \(2.5017\) | \(\Gamma_0(N)\)-optimal |
397800.dx4 | 397800dx4 | \([0, 0, 0, 4562925, -27738477250]\) | \(900753985478876/29018422265625\) | \(-338470877306250000000000\) | \([2]\) | \(37748736\) | \(3.1949\) |
Rank
sage: E.rank()
The elliptic curves in class 397800.dx have rank \(1\).
Complex multiplication
The elliptic curves in class 397800.dx do not have complex multiplication.Modular form 397800.2.a.dx
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.