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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 103488.u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 103488.u1 | 103488fm6 | \([0, -1, 0, -9263809, -10839151007]\) | \(5701568801608514/6277868289\) | \(96807803464261435392\) | \([2]\) | \(4718592\) | \(2.7498\) | |
| 103488.u2 | 103488fm4 | \([0, -1, 0, -726049, -76450751]\) | \(5489767279588/2847396321\) | \(21954122859762745344\) | \([2, 2]\) | \(2359296\) | \(2.4033\) | |
| 103488.u3 | 103488fm2 | \([0, -1, 0, -408529, 99772849]\) | \(3911877700432/38900169\) | \(74982457060245504\) | \([2, 2]\) | \(1179648\) | \(2.0567\) | |
| 103488.u4 | 103488fm1 | \([0, -1, 0, -407549, 100278333]\) | \(62140690757632/6237\) | \(751387456512\) | \([2]\) | \(589824\) | \(1.7101\) | \(\Gamma_0(N)\)-optimal |
| 103488.u5 | 103488fm3 | \([0, -1, 0, -106689, 243629793]\) | \(-17418812548/3314597517\) | \(-25556358481676402688\) | \([2]\) | \(2359296\) | \(2.4033\) | |
| 103488.u6 | 103488fm5 | \([0, -1, 0, 2731391, -597141215]\) | \(146142660369886/94532266521\) | \(-1457733844851638796288\) | \([2]\) | \(4718592\) | \(2.7498\) |
Rank
sage: E.rank()
The elliptic curves in class 103488.u have rank \(1\).
Complex multiplication
The elliptic curves in class 103488.u do not have complex multiplication.Modular form 103488.2.a.u
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
