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SageMath
E = EllipticCurve("cz1")
E.isogeny_class()
Elliptic curves in class 110400.cz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 110400.cz1 | 110400fl4 | \([0, -1, 0, -176640033, -903552768063]\) | \(148809678420065817601/20700\) | \(84787200000000\) | \([2]\) | \(7077888\) | \(2.9975\) | |
| 110400.cz2 | 110400fl6 | \([0, -1, 0, -41328033, 87610415937]\) | \(1905890658841300321/293666194803750\) | \(1202856733916160000000000\) | \([2]\) | \(14155776\) | \(3.3440\) | |
| 110400.cz3 | 110400fl3 | \([0, -1, 0, -11328033, -13339584063]\) | \(39248884582600321/3935264062500\) | \(16118841600000000000000\) | \([2, 2]\) | \(7077888\) | \(2.9975\) | |
| 110400.cz4 | 110400fl2 | \([0, -1, 0, -11040033, -14115168063]\) | \(36330796409313601/428490000\) | \(1755095040000000000\) | \([2, 2]\) | \(3538944\) | \(2.6509\) | |
| 110400.cz5 | 110400fl1 | \([0, -1, 0, -672033, -232416063]\) | \(-8194759433281/965779200\) | \(-3955831603200000000\) | \([2]\) | \(1769472\) | \(2.3043\) | \(\Gamma_0(N)\)-optimal |
| 110400.cz6 | 110400fl5 | \([0, -1, 0, 14063967, -64656816063]\) | \(75108181893694559/484313964843750\) | \(-1983750000000000000000000\) | \([2]\) | \(14155776\) | \(3.3440\) |
Rank
sage: E.rank()
The elliptic curves in class 110400.cz have rank \(0\).
Complex multiplication
The elliptic curves in class 110400.cz do not have complex multiplication.Modular form 110400.2.a.cz
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.
