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Properties

Label 82110.d
Number of curves $2$
Conductor $82110$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 82110.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
82110.d1 82110b2 \([1, 1, 0, -3584833, -2613965123]\) \(5094841618754165781496729/674331960579480\) \(674331960579480\) \([2]\) \(2242560\) \(2.2597\)  
82110.d2 82110b1 \([1, 1, 0, -223433, -41149563]\) \(-1233583919615644207129/14314037777726400\) \(-14314037777726400\) \([2]\) \(1121280\) \(1.9132\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 82110.d have rank \(1\).

Complex multiplication

The elliptic curves in class 82110.d do not have complex multiplication.

Modular form 82110.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{7} - q^{8} + q^{9} + q^{10} + 4 q^{11} - q^{12} - 6 q^{13} + q^{14} + q^{15} + q^{16} - q^{17} - q^{18} + 2 q^{19} + O(q^{20})\) 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.