Show commands:
SageMath
E = EllipticCurve("ir1")
E.isogeny_class()
Elliptic curves in class 127050.ir
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 127050.ir1 | 127050hz4 | \([1, 0, 0, -269863063313, -53958946691671383]\) | \(78519570041710065450485106721/96428056919040\) | \(2669190389742990960000000\) | \([2]\) | \(530841600\) | \(4.8611\) | |
| 127050.ir2 | 127050hz6 | \([1, 0, 0, -79371795313, 7877767739028617]\) | \(1997773216431678333214187041/187585177195046990066400\) | \(5192477876513041262015963287500000\) | \([2]\) | \(1061683200\) | \(5.2077\) | |
| 127050.ir3 | 127050hz3 | \([1, 0, 0, -17625495313, -763071229271383]\) | \(21876183941534093095979041/3572502915711058560000\) | \(98889169341562478337690000000000\) | \([2, 2]\) | \(530841600\) | \(4.8611\) | |
| 127050.ir4 | 127050hz2 | \([1, 0, 0, -16866583313, -843094705111383]\) | \(19170300594578891358373921/671785075055001600\) | \(18595441239836151398400000000\) | \([2, 2]\) | \(265420800\) | \(4.5145\) | |
| 127050.ir5 | 127050hz1 | \([1, 0, 0, -1006871313, -14408893399383]\) | \(-4078208988807294650401/880065599546327040\) | \(-24360779587467041832960000000\) | \([2]\) | \(132710400\) | \(4.1680\) | \(\Gamma_0(N)\)-optimal |
| 127050.ir6 | 127050hz5 | \([1, 0, 0, 31978212687, -4282404708163383]\) | \(130650216943167617311657439/361816948816603087500000\) | \(-10015324932226409098352929687500000\) | \([2]\) | \(1061683200\) | \(5.2077\) |
Rank
sage: E.rank()
The elliptic curves in class 127050.ir have rank \(1\).
Complex multiplication
The elliptic curves in class 127050.ir do not have complex multiplication.Modular form 127050.2.a.ir
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
