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SageMath
E = EllipticCurve("el1")
E.isogeny_class()
Elliptic curves in class 190575.el
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 190575.el1 | 190575ds6 | \([1, -1, 0, -123030342, -525219479559]\) | \(10206027697760497/5557167\) | \(112139112170059734375\) | \([2]\) | \(19660800\) | \(3.1758\) | |
| 190575.el2 | 190575ds4 | \([1, -1, 0, -7732467, -8108510184]\) | \(2533811507137/58110129\) | \(1172615160593093765625\) | \([2, 2]\) | \(9830400\) | \(2.8292\) | |
| 190575.el3 | 190575ds2 | \([1, -1, 0, -1062342, 235816191]\) | \(6570725617/2614689\) | \(52762298318697515625\) | \([2, 2]\) | \(4915200\) | \(2.4826\) | |
| 190575.el4 | 190575ds1 | \([1, -1, 0, -926217, 343218816]\) | \(4354703137/1617\) | \(32629745404265625\) | \([2]\) | \(2457600\) | \(2.1360\) | \(\Gamma_0(N)\)-optimal |
| 190575.el5 | 190575ds5 | \([1, -1, 0, 843408, -25114470309]\) | \(3288008303/13504609503\) | \(-272512040734023584484375\) | \([2]\) | \(19660800\) | \(3.1758\) | |
| 190575.el6 | 190575ds3 | \([1, -1, 0, 3429783, 1704741066]\) | \(221115865823/190238433\) | \(-3838856917066446515625\) | \([2]\) | \(9830400\) | \(2.8292\) |
Rank
sage: E.rank()
The elliptic curves in class 190575.el have rank \(0\).
Complex multiplication
The elliptic curves in class 190575.el do not have complex multiplication.Modular form 190575.2.a.el
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.
