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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 28322h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 28322.c5 | 28322h1 | \([1, 0, 1, -7376, -494070]\) | \(-15625/28\) | \(-79513303947868\) | \([2]\) | \(73728\) | \(1.3575\) | \(\Gamma_0(N)\)-optimal |
| 28322.c4 | 28322h2 | \([1, 0, 1, -148986, -22132078]\) | \(128787625/98\) | \(278296563817538\) | \([2]\) | \(147456\) | \(1.7041\) | |
| 28322.c6 | 28322h3 | \([1, 0, 1, 63429, 10324934]\) | \(9938375/21952\) | \(-62338430295128512\) | \([2]\) | \(221184\) | \(1.9068\) | |
| 28322.c3 | 28322h4 | \([1, 0, 1, -503011, 112510710]\) | \(4956477625/941192\) | \(2672760198903634952\) | \([2]\) | \(442368\) | \(2.2534\) | |
| 28322.c2 | 28322h5 | \([1, 0, 1, -2414746, 1448261196]\) | \(-548347731625/1835008\) | \(-5210983887527477248\) | \([2]\) | \(663552\) | \(2.4561\) | |
| 28322.c1 | 28322h6 | \([1, 0, 1, -38666906, 92542688844]\) | \(2251439055699625/25088\) | \(71243920337289728\) | \([2]\) | \(1327104\) | \(2.8027\) |
Rank
sage: E.rank()
The elliptic curves in class 28322h have rank \(0\).
Complex multiplication
The elliptic curves in class 28322h do not have complex multiplication.Modular form 28322.2.a.h
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
