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Properties

Label 176505.r
Number of curves $4$
Conductor $176505$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 176505.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
176505.r1 176505s4 \([1, 1, 0, -189147, -31739316]\) \(157551496201/13125\) \(62345118163125\) \([2]\) \(1105920\) \(1.6908\)  
176505.r2 176505s2 \([1, 1, 0, -12642, -427329]\) \(47045881/11025\) \(52369899257025\) \([2, 2]\) \(552960\) \(1.3442\)  
176505.r3 176505s1 \([1, 1, 0, -4237, 98824]\) \(1771561/105\) \(498760945305\) \([2]\) \(276480\) \(0.99768\) \(\Gamma_0(N)\)-optimal
176505.r4 176505s3 \([1, 1, 0, 29383, -2621034]\) \(590589719/972405\) \(-4619025114469605\) \([2]\) \(1105920\) \(1.6908\)  

Rank

sage: E.rank()
 

The elliptic curves in class 176505.r have rank \(0\).

Complex multiplication

The elliptic curves in class 176505.r do not have complex multiplication.

Modular form 176505.2.a.r

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.