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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 189618.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 189618.p1 | 189618ba3 | \([1, 0, 1, -2804481996, -57164824360694]\) | \(505384091400037554067434625/815656731648\) | \(3937019253229151232\) | \([2]\) | \(74649600\) | \(3.7234\) | |
| 189618.p2 | 189618ba4 | \([1, 0, 1, -2804454956, -57165981802486]\) | \(-505369473241574671219626625/20303219722982711328\) | \(-97999763687870457882392352\) | \([2]\) | \(149299200\) | \(4.0700\) | |
| 189618.p3 | 189618ba1 | \([1, 0, 1, -34720716, -77954209526]\) | \(959024269496848362625/11151660319506432\) | \(53826934395136521535488\) | \([2]\) | \(24883200\) | \(3.1741\) | \(\Gamma_0(N)\)-optimal |
| 189618.p4 | 189618ba2 | \([1, 0, 1, -7031756, -198855284470]\) | \(-7966267523043306625/3534510366354604032\) | \(-17060406446913699933093888\) | \([2]\) | \(49766400\) | \(3.5207\) |
Rank
sage: E.rank()
The elliptic curves in class 189618.p have rank \(0\).
Complex multiplication
The elliptic curves in class 189618.p do not have complex multiplication.Modular form 189618.2.a.p
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
