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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 140910b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 140910.dd3 | 140910b1 | \([1, 0, 0, -4823545, 3990649337]\) | \(12411455694361059171563281/301447494810000000000\) | \(301447494810000000000\) | \([10]\) | \(8960000\) | \(2.7139\) | \(\Gamma_0(N)\)-optimal |
| 140910.dd4 | 140910b2 | \([1, 0, 0, 676455, 12569549337]\) | \(34232640998190260436719/68272420831874513100000\) | \(-68272420831874513100000\) | \([10]\) | \(17920000\) | \(3.0604\) | |
| 140910.dd1 | 140910b3 | \([1, 0, 0, -368974045, -2727995360563]\) | \(5555346655662885019338595475281/46003971038489101892100\) | \(46003971038489101892100\) | \([2]\) | \(44800000\) | \(3.5186\) | |
| 140910.dd2 | 140910b4 | \([1, 0, 0, -360921495, -2852747075673]\) | \(-5199504460028149066396546564081/506641038664025374461800910\) | \(-506641038664025374461800910\) | \([2]\) | \(89600000\) | \(3.8652\) |
Rank
sage: E.rank()
The elliptic curves in class 140910b have rank \(1\).
Complex multiplication
The elliptic curves in class 140910b do not have complex multiplication.Modular form 140910.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
