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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 163170.ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 163170.ca1 | 163170de3 | \([1, -1, 0, -150383214, 709855836948]\) | \(4385367890843575421521/24975000000\) | \(2142008871975000000\) | \([2]\) | \(21233664\) | \(3.1291\) | |
| 163170.ca2 | 163170de6 | \([1, -1, 0, -133678134, -592273012500]\) | \(3080272010107543650001/15465841417699560\) | \(1326445226397232004606760\) | \([2]\) | \(42467328\) | \(3.4757\) | |
| 163170.ca3 | 163170de4 | \([1, -1, 0, -12932334, 2013665940]\) | \(2788936974993502801/1593609593601600\) | \(136677713231595651393600\) | \([2, 2]\) | \(21233664\) | \(3.1291\) | |
| 163170.ca4 | 163170de2 | \([1, -1, 0, -9404334, 11079920340]\) | \(1072487167529950801/2554882560000\) | \(219122366781749760000\) | \([2, 2]\) | \(10616832\) | \(2.7825\) | |
| 163170.ca5 | 163170de1 | \([1, -1, 0, -372654, 301513428]\) | \(-66730743078481/419010969600\) | \(-35936945519040921600\) | \([2]\) | \(5308416\) | \(2.4359\) | \(\Gamma_0(N)\)-optimal |
| 163170.ca6 | 163170de5 | \([1, -1, 0, 51365466, 16017726780]\) | \(174751791402194852399/102423900876336360\) | \(-8784500675851870288459560\) | \([2]\) | \(42467328\) | \(3.4757\) |
Rank
sage: E.rank()
The elliptic curves in class 163170.ca have rank \(1\).
Complex multiplication
The elliptic curves in class 163170.ca do not have complex multiplication.Modular form 163170.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
