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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 1110k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1110.k5 | 1110k1 | \([1, 1, 1, -845, 32195]\) | \(-66730743078481/419010969600\) | \(-419010969600\) | \([4]\) | \(1728\) | \(0.91367\) | \(\Gamma_0(N)\)-optimal |
| 1110.k4 | 1110k2 | \([1, 1, 1, -21325, 1187267]\) | \(1072487167529950801/2554882560000\) | \(2554882560000\) | \([2, 4]\) | \(3456\) | \(1.2602\) | |
| 1110.k3 | 1110k3 | \([1, 1, 1, -29325, 204867]\) | \(2788936974993502801/1593609593601600\) | \(1593609593601600\) | \([2, 2]\) | \(6912\) | \(1.6068\) | |
| 1110.k1 | 1110k4 | \([1, 1, 1, -341005, 76503875]\) | \(4385367890843575421521/24975000000\) | \(24975000000\) | \([4]\) | \(6912\) | \(1.6068\) | |
| 1110.k2 | 1110k5 | \([1, 1, 1, -303125, -64083373]\) | \(3080272010107543650001/15465841417699560\) | \(15465841417699560\) | \([2]\) | \(13824\) | \(1.9534\) | |
| 1110.k6 | 1110k6 | \([1, 1, 1, 116475, 1779507]\) | \(174751791402194852399/102423900876336360\) | \(-102423900876336360\) | \([2]\) | \(13824\) | \(1.9534\) |
Rank
sage: E.rank()
The elliptic curves in class 1110k have rank \(0\).
Complex multiplication
The elliptic curves in class 1110k do not have complex multiplication.Modular form 1110.2.a.k
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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
