Properties

Label 123627e
Number of curves $6$
Conductor $123627$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("123627.q1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 123627e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
123627.q5 123627e1 [1, 1, 0, -32308714, -75694250393] [2] 15482880 \(\Gamma_0(N)\)-optimal
123627.q4 123627e2 [1, 1, 0, -527022759, -4657043135520] [2, 2] 30965760  
123627.q3 123627e3 [1, 1, 0, -537118964, -4469344588365] [2, 2] 61931520  
123627.q1 123627e4 [1, 1, 0, -8432351274, -298041176049903] [2] 61931520  
123627.q6 123627e5 [1, 1, 0, 514328671, -19833307720512] [2] 123863040  
123627.q2 123627e6 [1, 1, 0, -1750105879, 22907527485802] [2] 123863040  

Rank

sage: E.rank()
 

The elliptic curves in class 123627e have rank \(0\).

Modular form 123627.2.a.q

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.