Properties

Label 51714p
Number of curves 6
Conductor 51714
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("51714.r1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 51714p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
51714.r5 51714p1 [1, -1, 1, -51746, -4188895] [2] 294912 \(\Gamma_0(N)\)-optimal
51714.r4 51714p2 [1, -1, 1, -173426, 22970081] [2, 2] 589824  
51714.r6 51714p3 [1, -1, 1, 343714, 133431185] [2] 1179648  
51714.r2 51714p4 [1, -1, 1, -2637446, 1649223281] [2, 2] 1179648  
51714.r3 51714p5 [1, -1, 1, -2500556, 1827946865] [2] 2359296  
51714.r1 51714p6 [1, -1, 1, -42198656, 105521136257] [2] 2359296  

Rank

sage: E.rank()
 

The elliptic curves in class 51714p have rank \(1\).

Modular form 51714.2.a.r

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} - 2q^{5} + q^{8} - 2q^{10} - 4q^{11} + q^{16} - q^{17} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.