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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 19166.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19166.a1 | 19166a6 | \([1, 0, 0, -3738083, -2782083455]\) | \(2251439055699625/25088\) | \(64368944148992\) | \([2]\) | \(311040\) | \(2.2186\) | |
| 19166.a2 | 19166a5 | \([1, 0, 0, -233443, -43557759]\) | \(-548347731625/1835008\) | \(-4708128486326272\) | \([2]\) | \(155520\) | \(1.8720\) | |
| 19166.a3 | 19166a4 | \([1, 0, 0, -48628, -3387192]\) | \(4956477625/941192\) | \(2414841170339528\) | \([2]\) | \(103680\) | \(1.6693\) | |
| 19166.a4 | 19166a2 | \([1, 0, 0, -14403, 663679]\) | \(128787625/98\) | \(251441188082\) | \([2]\) | \(34560\) | \(1.1199\) | |
| 19166.a5 | 19166a1 | \([1, 0, 0, -713, 14773]\) | \(-15625/28\) | \(-71840339452\) | \([2]\) | \(17280\) | \(0.77337\) | \(\Gamma_0(N)\)-optimal |
| 19166.a6 | 19166a3 | \([1, 0, 0, 6132, -309680]\) | \(9938375/21952\) | \(-56322826130368\) | \([2]\) | \(51840\) | \(1.3227\) |
Rank
sage: E.rank()
The elliptic curves in class 19166.a have rank \(1\).
Complex multiplication
The elliptic curves in class 19166.a do not have complex multiplication.Modular form 19166.2.a.a
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
