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Properties

Label 83370x
Number of curves $3$
Conductor $83370$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("x1")
 
E.isogeny_class()
 

Elliptic curves in class 83370x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
83370.x2 83370x1 \([1, 0, 0, -32007045, 750143318337]\) \(-3626272648660625410108067281/240996070457081856000000000\) \(-240996070457081856000000000\) \([9]\) \(39424320\) \(3.7420\) \(\Gamma_0(N)\)-optimal
83370.x3 83370x2 \([1, 0, 0, 287672955, -20100659817663]\) \(2632817860539621521875452252719/176067816626824667205829056000\) \(-176067816626824667205829056000\) \([3]\) \(118272960\) \(4.2913\)  
83370.x1 83370x3 \([1, 0, 0, -47924887245, -4039213143491223]\) \(-12173263182155064921392630443321792081/3458470272310068079827810615960\) \(-3458470272310068079827810615960\) \([]\) \(354818880\) \(4.8406\)  

Rank

sage: E.rank()
 

The elliptic curves in class 83370x have rank \(0\).

Complex multiplication

The elliptic curves in class 83370x do not have complex multiplication.

Modular form 83370.2.a.x

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10} + 3 q^{11} + q^{12} - 4 q^{13} + q^{14} + q^{15} + q^{16} + q^{18} + 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.