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SageMath
E = EllipticCurve("dx1")
E.isogeny_class()
Elliptic curves in class 205350.dx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 205350.dx1 | 205350v3 | \([1, 0, 0, -11670896838, 485292507728292]\) | \(4385367890843575421521/24975000000\) | \(1001234641637109375000000\) | \([2]\) | \(226934784\) | \(4.2170\) | |
| 205350.dx2 | 205350v6 | \([1, 0, 0, -10374453838, -404953052500708]\) | \(3080272010107543650001/15465841417699560\) | \(620017465043715017495000625000\) | \([2]\) | \(453869568\) | \(4.5636\) | |
| 205350.dx3 | 205350v4 | \([1, 0, 0, -1003648838, 1374423104292]\) | \(2788936974993502801/1593609593601600\) | \(63886972186552852260225000000\) | \([2, 2]\) | \(226934784\) | \(4.2170\) | |
| 205350.dx4 | 205350v2 | \([1, 0, 0, -729848838, 7573528904292]\) | \(1072487167529950801/2554882560000\) | \(102423900876336360000000000\) | \([2, 2]\) | \(113467392\) | \(3.8704\) | |
| 205350.dx5 | 205350v1 | \([1, 0, 0, -28920838, 206074696292]\) | \(-66730743078481/419010969600\) | \(-16797929849428377600000000\) | \([2]\) | \(56733696\) | \(3.5238\) | \(\Gamma_0(N)\)-optimal |
| 205350.dx6 | 205350v5 | \([1, 0, 0, 3986356162, 10960222709292]\) | \(174751791402194852399/102423900876336360\) | \(-4106120427987725656545800625000\) | \([2]\) | \(453869568\) | \(4.5636\) |
Rank
sage: E.rank()
The elliptic curves in class 205350.dx have rank \(1\).
Complex multiplication
The elliptic curves in class 205350.dx do not have complex multiplication.Modular form 205350.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
