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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 330330s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 330330.s6 | 330330s1 | \([1, 1, 0, -124075822, 949815100084]\) | \(-119241955571684699033281/151009623831306240000\) | \(-267522760204212713840640000\) | \([2]\) | \(141557760\) | \(3.7642\) | \(\Gamma_0(N)\)-optimal |
| 330330.s5 | 330330s2 | \([1, 1, 0, -2391673902, 44998814843316]\) | \(854030724380137731837009601/433088980424100000000\) | \(767243547249099020100000000\) | \([2, 2]\) | \(283115520\) | \(4.1108\) | |
| 330330.s2 | 330330s3 | \([1, 1, 0, -38262123902, 2880708543233316]\) | \(3496832439436665443750845809601/899932458715290000\) | \(1594285246494117867690000\) | \([2, 2]\) | \(566231040\) | \(4.4573\) | |
| 330330.s4 | 330330s4 | \([1, 1, 0, -2802793182, 28470421981764]\) | \(1374487385858093589979216321/597746248220214843750000\) | \(1058943941243252028808593750000\) | \([2]\) | \(566231040\) | \(4.4573\) | |
| 330330.s1 | 330330s5 | \([1, 1, 0, -612193982402, 184366112033324016]\) | \(14323025669278572844771245225753601/948647700\) | \(1680587268059700\) | \([2]\) | \(1132462080\) | \(4.8039\) | |
| 330330.s3 | 330330s6 | \([1, 1, 0, -38257465402, 2881445079102616]\) | \(-3495555353196166580084830665601/1773975660575132882567700\) | \(-3142706095224142984574517179700\) | \([2]\) | \(1132462080\) | \(4.8039\) |
Rank
sage: E.rank()
The elliptic curves in class 330330s have rank \(1\).
Complex multiplication
The elliptic curves in class 330330s do not have complex multiplication.Modular form 330330.2.a.s
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.
