Properties

Label 690.k
Number of curves $6$
Conductor $690$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 690.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
690.k1 690k4 \([1, 0, 0, -110400, 14109732]\) \(148809678420065817601/20700\) \(20700\) \([4]\) \(1536\) \(1.1530\)  
690.k2 690k5 \([1, 0, 0, -25830, -1370850]\) \(1905890658841300321/293666194803750\) \(293666194803750\) \([2]\) \(3072\) \(1.4996\)  
690.k3 690k3 \([1, 0, 0, -7080, 207900]\) \(39248884582600321/3935264062500\) \(3935264062500\) \([2, 2]\) \(1536\) \(1.1530\)  
690.k4 690k2 \([1, 0, 0, -6900, 220032]\) \(36330796409313601/428490000\) \(428490000\) \([2, 4]\) \(768\) \(0.80644\)  
690.k5 690k1 \([1, 0, 0, -420, 3600]\) \(-8194759433281/965779200\) \(-965779200\) \([8]\) \(384\) \(0.45987\) \(\Gamma_0(N)\)-optimal
690.k6 690k6 \([1, 0, 0, 8790, 1010922]\) \(75108181893694559/484313964843750\) \(-484313964843750\) \([2]\) \(3072\) \(1.4996\)  

Rank

sage: E.rank()
 

The elliptic curves in class 690.k have rank \(0\).

Complex multiplication

The elliptic curves in class 690.k do not have complex multiplication.

Modular form 690.2.a.k

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

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.