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Properties

Label 67158r
Number of curves $2$
Conductor $67158$
CM no
Rank $0$
Graph

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

Elliptic curves in class 67158r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
67158.q1 67158r1 \([1, -1, 0, -1402254, -638940204]\) \(-418288977642645996769/122877464621184\) \(-89577671708843136\) \([]\) \(1053696\) \(2.2319\) \(\Gamma_0(N)\)-optimal
67158.q2 67158r2 \([1, -1, 0, 7781256, 28217419146]\) \(71473535169369644529791/513262758348672548034\) \(-374168550836182287516786\) \([]\) \(7375872\) \(3.2049\)  

Rank

sage: E.rank()
 

The elliptic curves in class 67158r have rank \(0\).

Complex multiplication

The elliptic curves in class 67158r do not have complex multiplication.

Modular form 67158.2.a.r

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

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.