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Properties

Label 164730dk
Number of curves $4$
Conductor $164730$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("dk1")
 
E.isogeny_class()
 

Elliptic curves in class 164730dk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
164730.n4 164730dk1 \([1, 1, 0, 3607, -71163]\) \(214921799/218880\) \(-5283231102720\) \([2]\) \(589824\) \(1.1281\) \(\Gamma_0(N)\)-optimal
164730.n3 164730dk2 \([1, 1, 0, -19513, -676907]\) \(34043726521/11696400\) \(282322662051600\) \([2, 2]\) \(1179648\) \(1.4747\)  
164730.n2 164730dk3 \([1, 1, 0, -129333, 17355537]\) \(9912050027641/311647500\) \(7522413034927500\) \([2]\) \(2359296\) \(1.8213\)  
164730.n1 164730dk4 \([1, 1, 0, -279613, -57014567]\) \(100162392144121/23457780\) \(566213783336820\) \([2]\) \(2359296\) \(1.8213\)  

Rank

sage: E.rank()
 

The elliptic curves in class 164730dk have rank \(1\).

Complex multiplication

The elliptic curves in class 164730dk do not have complex multiplication.

Modular form 164730.2.a.dk

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.