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SageMath
E = EllipticCurve("bs1")
E.isogeny_class()
Elliptic curves in class 48510bs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 48510.bw5 | 48510bs1 | \([1, -1, 0, -146786859, 802080181125]\) | \(-4078208988807294650401/880065599546327040\) | \(-75479812698627830018211840\) | \([2]\) | \(17694720\) | \(3.6865\) | \(\Gamma_0(N)\)-optimal |
| 48510.bw4 | 48510bs2 | \([1, -1, 0, -2458896939, 46930063543173]\) | \(19170300594578891358373921/671785075055001600\) | \(57616400033161348880793600\) | \([2, 2]\) | \(35389440\) | \(4.0331\) | |
| 48510.bw3 | 48510bs3 | \([1, -1, 0, -2569535019, 42475708059525]\) | \(21876183941534093095979041/3572502915711058560000\) | \(306399717341727449495045760000\) | \([2, 2]\) | \(70778880\) | \(4.3797\) | |
| 48510.bw1 | 48510bs4 | \([1, -1, 0, -39342020139, 3003547608876933]\) | \(78519570041710065450485106721/96428056919040\) | \(8270260397513271843840\) | \([2]\) | \(70778880\) | \(4.3797\) | |
| 48510.bw6 | 48510bs5 | \([1, -1, 0, 4661947701, 238372236054693]\) | \(130650216943167617311657439/361816948816603087500000\) | \(-31031636212055587211498587500000\) | \([2]\) | \(141557760\) | \(4.7263\) | |
| 48510.bw2 | 48510bs6 | \([1, -1, 0, -11571227019, -438500898546075]\) | \(1997773216431678333214187041/187585177195046990066400\) | \(16088453005116840750720660434400\) | \([2]\) | \(141557760\) | \(4.7263\) |
Rank
sage: E.rank()
The elliptic curves in class 48510bs have rank \(1\).
Complex multiplication
The elliptic curves in class 48510bs do not have complex multiplication.Modular form 48510.2.a.bs
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
