Properties

Label 80850.a
Number of curves $2$
Conductor $80850$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 80850.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
80850.a1 80850p2 \([1, 1, 0, -9120150, -10602211500]\) \(45637459887836881/13417633152\) \(24665173792182000000\) \([2]\) \(5160960\) \(2.7000\)  
80850.a2 80850p1 \([1, 1, 0, -496150, -210291500]\) \(-7347774183121/6119866368\) \(-11249939973888000000\) \([2]\) \(2580480\) \(2.3534\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 80850.a have rank \(2\).

Complex multiplication

The elliptic curves in class 80850.a do not have complex multiplication.

Modular form 80850.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} - q^{11} - q^{12} - 6 q^{13} + q^{16} - 4 q^{17} - q^{18} + 2 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.