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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 164934.ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 164934.ca1 | 164934e3 | \([1, -1, 1, -699561176, 7121923869827]\) | \(441453577446719855661097/4354701912\) | \(373485891103523352\) | \([2]\) | \(33030144\) | \(3.4036\) | |
| 164934.ca2 | 164934e2 | \([1, -1, 1, -43723616, 111282688451]\) | \(107784459654566688937/10704361149504\) | \(918071533576059153984\) | \([2, 2]\) | \(16515072\) | \(3.0570\) | |
| 164934.ca3 | 164934e4 | \([1, -1, 1, -40424936, 128778887171]\) | \(-85183593440646799657/34223681512621656\) | \(-2935232409676971975716376\) | \([2]\) | \(33030144\) | \(3.4036\) | |
| 164934.ca4 | 164934e1 | \([1, -1, 1, -2939936, 1460394947]\) | \(32765849647039657/8229948198912\) | \(705850733051618660352\) | \([2]\) | \(8257536\) | \(2.7104\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 164934.ca have rank \(0\).
Complex multiplication
The elliptic curves in class 164934.ca do not have complex multiplication.Modular form 164934.2.a.ca
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.
