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Properties

Label 606f
Number of curves $2$
Conductor $606$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 606f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
606.f1 606f1 \([1, 0, 0, -90, 324]\) \(-80677568161/785376\) \(-785376\) \([5]\) \(100\) \(-0.048761\) \(\Gamma_0(N)\)-optimal
606.f2 606f2 \([1, 0, 0, 600, -10626]\) \(23885383766399/63060603006\) \(-63060603006\) \([]\) \(500\) \(0.75596\)  

Rank

sage: E.rank()
 

The elliptic curves in class 606f have rank \(0\).

Complex multiplication

The elliptic curves in class 606f do not have complex multiplication.

Modular form 606.2.a.f

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

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.