Properties

Label 80850gb
Number of curves $4$
Conductor $80850$
CM no
Rank $0$
Graph

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Show commands: 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)
 
\(q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} - q^{11} + q^{12} - 4 q^{13} + q^{16} - 6 q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.