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SageMath
E = EllipticCurve("gb1")
E.isogeny_class()
Elliptic curves in class 80850gb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
80850.fq3 | 80850gb1 | \([1, 0, 0, -6763, -202483]\) | \(18609625/1188\) | \(2183859562500\) | \([2]\) | \(207360\) | \(1.1183\) | \(\Gamma_0(N)\)-optimal |
80850.fq4 | 80850gb2 | \([1, 0, 0, 5487, -851733]\) | \(9938375/176418\) | \(-324303145031250\) | \([2]\) | \(414720\) | \(1.4649\) | |
80850.fq1 | 80850gb3 | \([1, 0, 0, -98638, 11869892]\) | \(57736239625/255552\) | \(469772457000000\) | \([2]\) | \(622080\) | \(1.6676\) | |
80850.fq2 | 80850gb4 | \([1, 0, 0, -49638, 23678892]\) | \(-7357983625/127552392\) | \(-234475177600125000\) | \([2]\) | \(1244160\) | \(2.0142\) |
Rank
sage: E.rank()
The elliptic curves in class 80850gb have rank \(0\).
Complex multiplication
The elliptic curves in class 80850gb do not have complex multiplication.Modular form 80850.2.a.gb
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.