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SageMath
E = EllipticCurve("eq1")
E.isogeny_class()
Elliptic curves in class 212160.eq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212160.eq1 | 212160fo6 | \([0, 1, 0, -144268801, -667018778785]\) | \(2533559197411478296569602/845325\) | \(110798438400\) | \([2]\) | \(12582912\) | \(2.8704\) | |
| 212160.eq2 | 212160fo4 | \([0, 1, 0, -9016801, -10424419585]\) | \(1237089966354690271204/714574355625\) | \(46830344970240000\) | \([2, 2]\) | \(6291456\) | \(2.5238\) | |
| 212160.eq3 | 212160fo5 | \([0, 1, 0, -8964801, -10550540385]\) | \(-607905111321334101602/14874581985380325\) | \(-1949641209987769958400\) | \([2]\) | \(12582912\) | \(2.8704\) | |
| 212160.eq4 | 212160fo3 | \([0, 1, 0, -1256321, 306582879]\) | \(3346154465291614084/1315155029296875\) | \(86190000000000000000\) | \([2]\) | \(6291456\) | \(2.5238\) | |
| 212160.eq5 | 212160fo2 | \([0, 1, 0, -566801, -161049585]\) | \(1229125878116884816/29018422265625\) | \(475437830400000000\) | \([2, 2]\) | \(3145728\) | \(2.1772\) | |
| 212160.eq6 | 212160fo1 | \([0, 1, 0, 4419, -7848381]\) | \(9317458724864/26001416731875\) | \(-26625450733440000\) | \([2]\) | \(1572864\) | \(1.8307\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 212160.eq have rank \(1\).
Complex multiplication
The elliptic curves in class 212160.eq do not have complex multiplication.Modular form 212160.2.a.eq
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
