Properties

Label 379050kd
Number of curves $6$
Conductor $379050$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("kd1")
 
E.isogeny_class()
 

Elliptic curves in class 379050kd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
379050.kd4 379050kd1 \([1, 0, 0, -91197813, -335223292383]\) \(114113060120923921/124104960\) \(91228549682340000000\) \([2]\) \(66355200\) \(3.1174\) \(\Gamma_0(N)\)-optimal
379050.kd3 379050kd2 \([1, 0, 0, -91919813, -329645842383]\) \(116844823575501841/3760263939600\) \(2764139528609574806250000\) \([2, 2]\) \(132710400\) \(3.4640\)  
379050.kd2 379050kd3 \([1, 0, 0, -223504313, 826850328117]\) \(1679731262160129361/570261564022500\) \(419194651248069005039062500\) \([2, 2]\) \(265420800\) \(3.8105\)  
379050.kd5 379050kd4 \([1, 0, 0, 28112687, -1129182324883]\) \(3342636501165359/751262567039460\) \(-552247020760827460379062500\) \([2]\) \(265420800\) \(3.8105\)  
379050.kd1 379050kd5 \([1, 0, 0, -3208523063, 69938989446867]\) \(4969327007303723277361/1123462695162150\) \(825848316633402884439843750\) \([2]\) \(530841600\) \(4.1571\)  
379050.kd6 379050kd6 \([1, 0, 0, 656162437, 5730992459367]\) \(42502666283088696719/43898058864843750\) \(-32269107085725533532714843750\) \([2]\) \(530841600\) \(4.1571\)  

Rank

sage: E.rank()
 

The elliptic curves in class 379050kd have rank \(1\).

Complex multiplication

The elliptic curves in class 379050kd do not have complex multiplication.

Modular form 379050.2.a.kd

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + q^{8} + q^{9} + 4 q^{11} + q^{12} + 6 q^{13} + q^{14} + q^{16} - 2 q^{17} + q^{18} + O(q^{20})\) Copy content Toggle raw display

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.