Properties

Label 35378p
Number of curves 6
Conductor 35378
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("35378.k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 35378p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
35378.k5 35378p1 [1, 0, 0, -9213, 688141] [2] 114048 \(\Gamma_0(N)\)-optimal
35378.k4 35378p2 [1, 0, 0, -186103, 30865575] [2] 228096  
35378.k6 35378p3 [1, 0, 0, 79232, -14400576] [2] 342144  
35378.k3 35378p4 [1, 0, 0, -628328, -157186184] [2] 684288  
35378.k2 35378p5 [1, 0, 0, -3016343, -2022438167] [2] 1026432  
35378.k1 35378p6 [1, 0, 0, -48300183, -129206631191] [2] 2052864  

Rank

sage: E.rank()
 

The elliptic curves in class 35378p have rank \(0\).

Modular form 35378.2.a.k

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.