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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 117117.l
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 117117.l1 | 117117r6 | \([1, -1, 1, -6873431, 6937704960]\) | \(10206027697760497/5557167\) | \(19554246710085087\) | \([2]\) | \(2457600\) | \(2.4546\) | |
| 117117.l2 | 117117r4 | \([1, -1, 1, -431996, 107207286]\) | \(2533811507137/58110129\) | \(204474653869655169\) | \([2, 2]\) | \(1228800\) | \(2.1080\) | |
| 117117.l3 | 117117r2 | \([1, -1, 1, -59351, -3095634]\) | \(6570725617/2614689\) | \(9200420605705329\) | \([2, 2]\) | \(614400\) | \(1.7614\) | |
| 117117.l4 | 117117r1 | \([1, -1, 1, -51746, -4516248]\) | \(4354703137/1617\) | \(5689808661537\) | \([2]\) | \(307200\) | \(1.4149\) | \(\Gamma_0(N)\)-optimal |
| 117117.l5 | 117117r5 | \([1, -1, 1, 47119, 331624752]\) | \(3288008303/13504609503\) | \(-47519260433422560783\) | \([2]\) | \(2457600\) | \(2.4546\) | |
| 117117.l6 | 117117r3 | \([1, -1, 1, 191614, -22570518]\) | \(221115865823/190238433\) | \(-669400299221166513\) | \([2]\) | \(1228800\) | \(2.1080\) |
Rank
sage: E.rank()
The elliptic curves in class 117117.l have rank \(2\).
Complex multiplication
The elliptic curves in class 117117.l do not have complex multiplication.Modular form 117117.2.a.l
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
