Properties

Label 188598x
Number of curves 6
Conductor 188598
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("188598.c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 188598x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
188598.c5 188598x1 [1, 1, 0, -62904, -5658048] [2] 1204224 \(\Gamma_0(N)\)-optimal
188598.c4 188598x2 [1, 1, 0, -210824, 30641520] [2, 2] 2408448  
188598.c2 188598x3 [1, 1, 0, -3206204, 2208282780] [2, 2] 4816896  
188598.c6 188598x4 [1, 1, 0, 417836, 179131012] [2] 4816896  
188598.c1 188598x5 [1, 1, 0, -51298694, 141397567338] [2] 9633792  
188598.c3 188598x6 [1, 1, 0, -3039794, 2447946462] [2] 9633792  

Rank

sage: E.rank()
 

The elliptic curves in class 188598x have rank \(1\).

Modular form 188598.2.a.c

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

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.