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SageMath
E = EllipticCurve("ch1")
E.isogeny_class()
Elliptic curves in class 182070ch
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 182070.cg3 | 182070ch1 | \([1, -1, 1, -13493, 531981]\) | \(416832723/56000\) | \(36496004328000\) | \([2]\) | \(497664\) | \(1.3298\) | \(\Gamma_0(N)\)-optimal |
| 182070.cg4 | 182070ch2 | \([1, -1, 1, 21187, 2793117]\) | \(1613964717/6125000\) | \(-3991750473375000\) | \([2]\) | \(995328\) | \(1.6764\) | |
| 182070.cg1 | 182070ch3 | \([1, -1, 1, -273593, -54944459]\) | \(4767078987/6860\) | \(3259184426501220\) | \([2]\) | \(1492992\) | \(1.8791\) | |
| 182070.cg2 | 182070ch4 | \([1, -1, 1, -195563, -86999183]\) | \(-1740992427/5882450\) | \(-2794750645724796150\) | \([2]\) | \(2985984\) | \(2.2257\) |
Rank
sage: E.rank()
The elliptic curves in class 182070ch have rank \(0\).
Complex multiplication
The elliptic curves in class 182070ch do not have complex multiplication.Modular form 182070.2.a.ch
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
