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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 55440.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55440.j1 | 55440cs4 | \([0, 0, 0, -12846373923, 560426873165602]\) | \(78519570041710065450485106721/96428056919040\) | \(287932635111342735360\) | \([2]\) | \(35389440\) | \(4.0999\) | |
| 55440.j2 | 55440cs6 | \([0, 0, 0, -3778359843, -81819949406942]\) | \(1997773216431678333214187041/187585177195046990066400\) | \(560126337741575191586429337600\) | \([2]\) | \(70778880\) | \(4.4465\) | |
| 55440.j3 | 55440cs3 | \([0, 0, 0, -839031843, 7925376685858]\) | \(21876183941534093095979041/3572502915711058560000\) | \(10667436546266569483223040000\) | \([2, 2]\) | \(35389440\) | \(4.0999\) | |
| 55440.j4 | 55440cs2 | \([0, 0, 0, -802905123, 8756515231522]\) | \(19170300594578891358373921/671785075055001600\) | \(2005939485553033897574400\) | \([2, 2]\) | \(17694720\) | \(3.7533\) | |
| 55440.j5 | 55440cs1 | \([0, 0, 0, -47930403, 149652428578]\) | \(-4078208988807294650401/880065599546327040\) | \(-2627861799195739800207360\) | \([2]\) | \(8847360\) | \(3.4067\) | \(\Gamma_0(N)\)-optimal |
| 55440.j6 | 55440cs5 | \([0, 0, 0, 1522268637, 44477835856162]\) | \(130650216943167617311657439/361816948816603087500000\) | \(-1080379620095195753625600000000\) | \([2]\) | \(70778880\) | \(4.4465\) |
Rank
sage: E.rank()
The elliptic curves in class 55440.j have rank \(1\).
Complex multiplication
The elliptic curves in class 55440.j do not have complex multiplication.Modular form 55440.2.a.j
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.
