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Properties

Label 174570ca
Number of curves $2$
Conductor $174570$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ca1")
 
E.isogeny_class()
 

Elliptic curves in class 174570ca

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
174570.z2 174570ca1 \([1, 0, 1, 48921, 3428566]\) \(165348311/160380\) \(-12559515819366780\) \([3]\) \(1271808\) \(1.7762\) \(\Gamma_0(N)\)-optimal
174570.z1 174570ca2 \([1, 0, 1, -498594, -189077708]\) \(-175041455449/95832000\) \(-7504698341448792000\) \([]\) \(3815424\) \(2.3255\)  

Rank

sage: E.rank()
 

The elliptic curves in class 174570ca have rank \(1\).

Complex multiplication

The elliptic curves in class 174570ca do not have complex multiplication.

Modular form 174570.2.a.ca

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.