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SageMath
E = EllipticCurve("hl1")
E.isogeny_class()
Elliptic curves in class 212160hl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212160.x4 | 212160hl1 | \([0, -1, 0, -66301, 6593101]\) | \(31476797652269056/49725\) | \(50918400\) | \([2]\) | \(360448\) | \(1.1732\) | \(\Gamma_0(N)\)-optimal |
| 212160.x3 | 212160hl2 | \([0, -1, 0, -66321, 6588945]\) | \(1969080716416336/2472575625\) | \(40510679040000\) | \([2, 2]\) | \(720896\) | \(1.5198\) | |
| 212160.x5 | 212160hl3 | \([0, -1, 0, -48641, 10163841]\) | \(-194204905090564/566398828125\) | \(-37119513600000000\) | \([2]\) | \(1441792\) | \(1.8664\) | |
| 212160.x2 | 212160hl4 | \([0, -1, 0, -84321, 2747745]\) | \(1011710313226084/536724738225\) | \(35174792444313600\) | \([2, 2]\) | \(1441792\) | \(1.8664\) | |
| 212160.x6 | 212160hl5 | \([0, -1, 0, 321279, 21161985]\) | \(27980756504588158/17683545112935\) | \(-2317817625042616320\) | \([2]\) | \(2883584\) | \(2.2129\) | |
| 212160.x1 | 212160hl6 | \([0, -1, 0, -777921, -261791295]\) | \(397210600760070242/3536192675535\) | \(463495846367723520\) | \([2]\) | \(2883584\) | \(2.2129\) |
Rank
sage: E.rank()
The elliptic curves in class 212160hl have rank \(2\).
Complex multiplication
The elliptic curves in class 212160hl do not have complex multiplication.Modular form 212160.2.a.hl
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
