Properties

Label 60690p
Number of curves $4$
Conductor $60690$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 60690p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
60690.p4 60690p1 \([1, 1, 0, 9098, 4865824]\) \(3449795831/425079900\) \(-10260395416763100\) \([2]\) \(663552\) \(1.7516\) \(\Gamma_0(N)\)-optimal
60690.p2 60690p2 \([1, 1, 0, -381052, 87499594]\) \(253503932606569/9180151470\) \(221586539537576430\) \([2]\) \(1327104\) \(2.0982\)  
60690.p3 60690p3 \([1, 1, 0, -81937, -131850539]\) \(-2520453225529/309519000000\) \(-7471036219311000000\) \([2]\) \(1990656\) \(2.3009\)  
60690.p1 60690p4 \([1, 1, 0, -4416937, -3546963539]\) \(394815279796185529/3548222643000\) \(85645468872774867000\) \([2]\) \(3981312\) \(2.6475\)  

Rank

sage: E.rank()
 

The elliptic curves in class 60690p have rank \(0\).

Complex multiplication

The elliptic curves in class 60690p do not have complex multiplication.

Modular form 60690.2.a.p

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.