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SageMath
E = EllipticCurve("cy1")
E.isogeny_class()
Elliptic curves in class 227850.cy
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 227850.cy1 | 227850ga5 | \([1, 0, 1, -376712026, 2814216436448]\) | \(3216206300355197383681/57660\) | \(105994395937500\) | \([2]\) | \(37748736\) | \(3.1595\) | |
| 227850.cy2 | 227850ga3 | \([1, 0, 1, -23544526, 43970566448]\) | \(785209010066844481/3324675600\) | \(6111636869756250000\) | \([2, 2]\) | \(18874368\) | \(2.8129\) | |
| 227850.cy3 | 227850ga6 | \([1, 0, 1, -23177026, 45409696448]\) | \(-749011598724977281/51173462246460\) | \(-94070416559902695937500\) | \([2]\) | \(37748736\) | \(3.1595\) | |
| 227850.cy4 | 227850ga4 | \([1, 0, 1, -4532526, -2896169552]\) | \(5601911201812801/1271193750000\) | \(2336791773339843750000\) | \([2]\) | \(18874368\) | \(2.8129\) | |
| 227850.cy5 | 227850ga2 | \([1, 0, 1, -1494526, 664366448]\) | \(200828550012481/12454560000\) | \(22894789522500000000\) | \([2, 2]\) | \(9437184\) | \(2.4664\) | |
| 227850.cy6 | 227850ga1 | \([1, 0, 1, 73474, 43438448]\) | \(23862997439/457113600\) | \(-840296217600000000\) | \([2]\) | \(4718592\) | \(2.1198\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 227850.cy have rank \(0\).
Complex multiplication
The elliptic curves in class 227850.cy do not have complex multiplication.Modular form 227850.2.a.cy
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
