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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 23232.cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 23232.cp1 | 23232cf4 | \([0, 1, 0, -151798049, 719808335871]\) | \(6663712298552914184/29403\) | \(1706859170463744\) | \([2]\) | \(1843200\) | \(3.0093\) | |
| 23232.cp2 | 23232cf2 | \([0, 1, 0, -9487529, 11244256791]\) | \(13015685560572352/864536409\) | \(6273347523643183104\) | \([2, 2]\) | \(921600\) | \(2.6627\) | |
| 23232.cp3 | 23232cf3 | \([0, 1, 0, -8901889, 12693481535]\) | \(-1343891598641864/421900912521\) | \(-24491563499081409527808\) | \([2]\) | \(1843200\) | \(3.0093\) | |
| 23232.cp4 | 23232cf1 | \([0, 1, 0, -629724, 152513370]\) | \(243578556889408/52089208083\) | \(5905869411886564032\) | \([2]\) | \(460800\) | \(2.3162\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 23232.cp have rank \(1\).
Complex multiplication
The elliptic curves in class 23232.cp do not have complex multiplication.Modular form 23232.2.a.cp
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.
