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SageMath
E = EllipticCurve("hy1")
E.isogeny_class()
Elliptic curves in class 436800.hy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
436800.hy1 | 436800hy7 | \([0, -1, 0, -64304728033, 6276448294455937]\) | \(7179471593960193209684686321/49441793310\) | \(202513585397760000000\) | \([2]\) | \(509607936\) | \(4.4322\) | \(\Gamma_0(N)\)-optimal* |
436800.hy2 | 436800hy6 | \([0, -1, 0, -4019048033, 98070379655937]\) | \(1752803993935029634719121/4599740941532100\) | \(18840538896515481600000000\) | \([2, 2]\) | \(254803968\) | \(4.0856\) | \(\Gamma_0(N)\)-optimal* |
436800.hy3 | 436800hy8 | \([0, -1, 0, -3969656033, 100598212823937]\) | \(-1688971789881664420008241/89901485966373558750\) | \(-368236486518266096640000000000\) | \([2]\) | \(509607936\) | \(4.4322\) | |
436800.hy4 | 436800hy4 | \([0, -1, 0, -794248033, 8601647655937]\) | \(13527956825588849127121/25701087819771000\) | \(105271655709782016000000000\) | \([2]\) | \(169869312\) | \(3.8829\) | \(\Gamma_0(N)\)-optimal* |
436800.hy5 | 436800hy3 | \([0, -1, 0, -254280033, 1492786151937]\) | \(443915739051786565201/21894701746029840\) | \(89680698351738224640000000\) | \([2]\) | \(127401984\) | \(3.7391\) | \(\Gamma_0(N)\)-optimal* |
436800.hy6 | 436800hy2 | \([0, -1, 0, -66248033, 36727655937]\) | \(7850236389974007121/4400862921000000\) | \(18025934524416000000000000\) | \([2, 2]\) | \(84934656\) | \(3.5363\) | \(\Gamma_0(N)\)-optimal* |
436800.hy7 | 436800hy1 | \([0, -1, 0, -41160033, -101080728063]\) | \(1882742462388824401/11650189824000\) | \(47719177519104000000000\) | \([2]\) | \(42467328\) | \(3.1898\) | \(\Gamma_0(N)\)-optimal* |
436800.hy8 | 436800hy5 | \([0, -1, 0, 260343967, 291142823937]\) | \(476437916651992691759/284661685546875000\) | \(-1165974264000000000000000000\) | \([2]\) | \(169869312\) | \(3.8829\) |
Rank
sage: E.rank()
The elliptic curves in class 436800.hy have rank \(2\).
Complex multiplication
The elliptic curves in class 436800.hy do not have complex multiplication.Modular form 436800.2.a.hy
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.