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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 46410be
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 46410.be7 | 46410be1 | \([1, 0, 1, -63473, -4407244]\) | \(28280100765151839241/7994847656250000\) | \(7994847656250000\) | \([6]\) | \(304128\) | \(1.7583\) | \(\Gamma_0(N)\)-optimal |
| 46410.be6 | 46410be2 | \([1, 0, 1, -375973, 85217756]\) | \(5877491705974396839241/261806444735062500\) | \(261806444735062500\) | \([2, 6]\) | \(608256\) | \(2.1049\) | |
| 46410.be4 | 46410be3 | \([1, 0, 1, -4722848, -3950910994]\) | \(11650256451486052494789241/580277967360000\) | \(580277967360000\) | \([2]\) | \(912384\) | \(2.3076\) | |
| 46410.be8 | 46410be4 | \([1, 0, 1, 197777, 322291256]\) | \(855567391070976980759/45363085180055574750\) | \(-45363085180055574750\) | \([6]\) | \(1216512\) | \(2.4515\) | |
| 46410.be2 | 46410be5 | \([1, 0, 1, -5949723, 5585394256]\) | \(23292378980986805290659241/49479832772574750\) | \(49479832772574750\) | \([6]\) | \(1216512\) | \(2.4515\) | |
| 46410.be3 | 46410be6 | \([1, 0, 1, -4730848, -3936856594]\) | \(11709559667189768059461241/82207646338733697600\) | \(82207646338733697600\) | \([2, 2]\) | \(1824768\) | \(2.6542\) | |
| 46410.be5 | 46410be7 | \([1, 0, 1, -1783048, -8785398034]\) | \(-626920492174472718626041/32979221374608565962360\) | \(-32979221374608565962360\) | \([2]\) | \(3649536\) | \(3.0008\) | |
| 46410.be1 | 46410be8 | \([1, 0, 1, -7806648, 1811198446]\) | \(52615951054626272117608441/29030877531795041917560\) | \(29030877531795041917560\) | \([2]\) | \(3649536\) | \(3.0008\) |
Rank
sage: E.rank()
The elliptic curves in class 46410be have rank \(1\).
Complex multiplication
The elliptic curves in class 46410be do not have complex multiplication.Modular form 46410.2.a.be
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
