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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 2142.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2142.h1 | 2142g5 | \([1, -1, 0, -123467436, 528082919464]\) | \(285531136548675601769470657/17941034271597192\) | \(13079013983994352968\) | \([2]\) | \(245760\) | \(3.1284\) | |
2142.h2 | 2142g3 | \([1, -1, 0, -7731396, 8219774992]\) | \(70108386184777836280897/552468975892674624\) | \(402749883425759800896\) | \([2, 2]\) | \(122880\) | \(2.7819\) | |
2142.h3 | 2142g6 | \([1, -1, 0, -2633436, 18893883640]\) | \(-2770540998624539614657/209924951154647363208\) | \(-153035289391737927778632\) | \([2]\) | \(245760\) | \(3.1284\) | |
2142.h4 | 2142g2 | \([1, -1, 0, -816516, -71166128]\) | \(82582985847542515777/44772582831427584\) | \(32639212884110708736\) | \([2, 2]\) | \(61440\) | \(2.4353\) | |
2142.h5 | 2142g1 | \([1, -1, 0, -632196, -193075376]\) | \(38331145780597164097/55468445663232\) | \(40436496888496128\) | \([2]\) | \(30720\) | \(2.0887\) | \(\Gamma_0(N)\)-optimal |
2142.h6 | 2142g4 | \([1, -1, 0, 3149244, -562127216]\) | \(4738217997934888496063/2928751705237796928\) | \(-2135059993118353960512\) | \([2]\) | \(122880\) | \(2.7819\) |
Rank
sage: E.rank()
The elliptic curves in class 2142.h have rank \(1\).
Complex multiplication
The elliptic curves in class 2142.h do not have complex multiplication.Modular form 2142.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.