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SageMath
E = EllipticCurve("hp1")
E.isogeny_class()
Elliptic curves in class 92400.hp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 92400.hp1 | 92400hi8 | \([0, 1, 0, -102571008, -375179112012]\) | \(1864737106103260904761/129177711985836360\) | \(8267373567093527040000000\) | \([2]\) | \(15925248\) | \(3.5291\) | |
| 92400.hp2 | 92400hi5 | \([0, 1, 0, -100801008, -389567652012]\) | \(1769857772964702379561/691787250\) | \(44274384000000000\) | \([2]\) | \(5308416\) | \(2.9798\) | |
| 92400.hp3 | 92400hi6 | \([0, 1, 0, -20251008, 28024247988]\) | \(14351050585434661561/3001282273281600\) | \(192082065490022400000000\) | \([2, 2]\) | \(7962624\) | \(3.1826\) | |
| 92400.hp4 | 92400hi3 | \([0, 1, 0, -19099008, 32118455988]\) | \(12038605770121350841/757333463040\) | \(48469341634560000000\) | \([2]\) | \(3981312\) | \(2.8360\) | |
| 92400.hp5 | 92400hi2 | \([0, 1, 0, -6301008, -6086652012]\) | \(432288716775559561/270140062500\) | \(17288964000000000000\) | \([2, 2]\) | \(2654208\) | \(2.6333\) | |
| 92400.hp6 | 92400hi4 | \([0, 1, 0, -5113008, -8450772012]\) | \(-230979395175477481/348191894531250\) | \(-22284281250000000000000\) | \([2]\) | \(5308416\) | \(2.9798\) | |
| 92400.hp7 | 92400hi1 | \([0, 1, 0, -469008, -56364012]\) | \(178272935636041/81841914000\) | \(5237882496000000000\) | \([2]\) | \(1327104\) | \(2.2867\) | \(\Gamma_0(N)\)-optimal |
| 92400.hp8 | 92400hi7 | \([0, 1, 0, 43636992, 169216727988]\) | \(143584693754978072519/276341298967965000\) | \(-17685843133949760000000000\) | \([2]\) | \(15925248\) | \(3.5291\) |
Rank
sage: E.rank()
The elliptic curves in class 92400.hp have rank \(1\).
Complex multiplication
The elliptic curves in class 92400.hp do not have complex multiplication.Modular form 92400.2.a.hp
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
