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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 122034b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 122034.b4 | 122034b1 | \([1, 1, 0, -13440419, -18962513907]\) | \(42476766863084497/22597926912\) | \(142849700165519474688\) | \([2]\) | \(5677056\) | \(2.8171\) | \(\Gamma_0(N)\)-optimal |
| 122034.b3 | 122034b2 | \([1, 1, 0, -15807139, -11827799795]\) | \(69099171213056977/30438083453184\) | \(192410176023335658998016\) | \([2, 2]\) | \(11354112\) | \(3.1636\) | |
| 122034.b6 | 122034b3 | \([1, 1, 0, 54159021, -87964975107]\) | \(2779235713829366063/2137426732088208\) | \(-13511450364167220659826192\) | \([2]\) | \(22708224\) | \(3.5102\) | |
| 122034.b2 | 122034b4 | \([1, 1, 0, -123640819, 520978413085]\) | \(33067289963089830097/583837617703824\) | \(3690649543169141259599376\) | \([2, 2]\) | \(22708224\) | \(3.5102\) | |
| 122034.b5 | 122034b5 | \([1, 1, 0, -2827159, 1495002302737]\) | \(-395333776029457/152741401227689868\) | \(-965533849773201967206887532\) | \([2]\) | \(45416448\) | \(3.8568\) | |
| 122034.b1 | 122034b6 | \([1, 1, 0, -1969793359, 33648708821353]\) | \(133713416563031354522257/42805753664652\) | \(270590709500327490243948\) | \([2]\) | \(45416448\) | \(3.8568\) |
Rank
sage: E.rank()
The elliptic curves in class 122034b have rank \(0\).
Complex multiplication
The elliptic curves in class 122034b do not have complex multiplication.Modular form 122034.2.a.b
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
