Properties

Label 60690.bp
Number of curves $8$
Conductor $60690$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("60690.bp1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 60690.bp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
60690.bp1 60690bl8 [1, 1, 1, -1864345, 615650345] [2] 2654208  
60690.bp2 60690bl5 [1, 1, 1, -1664935, 826189157] [2] 884736  
60690.bp3 60690bl6 [1, 1, 1, -780595, -258719155] [2, 2] 1327104  
60690.bp4 60690bl3 [1, 1, 1, -774815, -262832203] [2] 663552  
60690.bp5 60690bl2 [1, 1, 1, -104335, 12804437] [2, 2] 442368  
60690.bp6 60690bl4 [1, 1, 1, -23415, 32257605] [2] 884736  
60690.bp7 60690bl1 [1, 1, 1, -11855, -179755] [2] 221184 \(\Gamma_0(N)\)-optimal
60690.bp8 60690bl7 [1, 1, 1, 210675, -869737983] [2] 2654208  

Rank

sage: E.rank()
 

The elliptic curves in class 60690.bp have rank \(1\).

Modular form 60690.2.a.bp

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

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 LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.