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SageMath
E = EllipticCurve("gr1")
E.isogeny_class()
Elliptic curves in class 127050.gr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 127050.gr1 | 127050gb8 | \([1, 1, 1, -19514338, -20882966719]\) | \(29689921233686449/10380965400750\) | \(287351772598719855468750\) | \([2]\) | \(19906560\) | \(3.2027\) | |
| 127050.gr2 | 127050gb5 | \([1, 1, 1, -17427088, -28009019719]\) | \(21145699168383889/2593080\) | \(71778115591875000\) | \([2]\) | \(6635520\) | \(2.6534\) | |
| 127050.gr3 | 127050gb6 | \([1, 1, 1, -8170588, 8746908281]\) | \(2179252305146449/66177562500\) | \(1831837325000976562500\) | \([2, 2]\) | \(9953280\) | \(2.8561\) | |
| 127050.gr4 | 127050gb3 | \([1, 1, 1, -8110088, 8886300281]\) | \(2131200347946769/2058000\) | \(56966758406250000\) | \([2]\) | \(4976640\) | \(2.5095\) | |
| 127050.gr5 | 127050gb2 | \([1, 1, 1, -1092088, -435539719]\) | \(5203798902289/57153600\) | \(1582048262025000000\) | \([2, 2]\) | \(3317760\) | \(2.3068\) | |
| 127050.gr6 | 127050gb4 | \([1, 1, 1, -245088, -1092811719]\) | \(-58818484369/18600435000\) | \(-514871956703671875000\) | \([2]\) | \(6635520\) | \(2.6534\) | |
| 127050.gr7 | 127050gb1 | \([1, 1, 1, -124088, 5868281]\) | \(7633736209/3870720\) | \(107144009280000000\) | \([2]\) | \(1658880\) | \(1.9602\) | \(\Gamma_0(N)\)-optimal |
| 127050.gr8 | 127050gb7 | \([1, 1, 1, 2205162, 29456905281]\) | \(42841933504271/13565917968750\) | \(-375513300041198730468750\) | \([2]\) | \(19906560\) | \(3.2027\) |
Rank
sage: E.rank()
The elliptic curves in class 127050.gr have rank \(0\).
Complex multiplication
The elliptic curves in class 127050.gr do not have complex multiplication.Modular form 127050.2.a.gr
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.
