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SageMath
E = EllipticCurve("gv1")
E.isogeny_class()
Elliptic curves in class 450800.gv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
450800.gv1 | 450800gv4 | \([0, -1, 0, -3269888008, -71967999865488]\) | \(513516182162686336369/1944885031250\) | \(14644081858658000000000000\) | \([2]\) | \(414056448\) | \(4.0434\) | |
450800.gv2 | 450800gv3 | \([0, -1, 0, -207388008, -1089499865488]\) | \(131010595463836369/7704101562500\) | \(58008310062500000000000000\) | \([2]\) | \(207028224\) | \(3.6969\) | |
450800.gv3 | 450800gv2 | \([0, -1, 0, -55684008, -17110169488]\) | \(2535986675931409/1450751712200\) | \(10923487244071539200000000\) | \([2]\) | \(138018816\) | \(3.4941\) | |
450800.gv4 | 450800gv1 | \([0, -1, 0, -36084008, 83085030512]\) | \(690080604747409/3406760000\) | \(25651322063360000000000\) | \([2]\) | \(69009408\) | \(3.1476\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 450800.gv have rank \(0\).
Complex multiplication
The elliptic curves in class 450800.gv do not have complex multiplication.Modular form 450800.2.a.gv
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.