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Properties

Label 430950.hp
Number of curves $8$
Conductor $430950$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("hp1")
 
E.isogeny_class()
 

Elliptic curves in class 430950.hp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
430950.hp1 430950hp8 \([1, 0, 0, -479410838, -4040285504958]\) \(161572377633716256481/914742821250\) \(68988888785857675781250\) \([2]\) \(113246208\) \(3.5739\)  
430950.hp2 430950hp3 \([1, 0, 0, -91936088, 339287091792]\) \(1139466686381936641/4080\) \(307709073750000\) \([2]\) \(28311552\) \(2.8808\) \(\Gamma_0(N)\)-optimal*
430950.hp3 430950hp6 \([1, 0, 0, -30504588, -60731598708]\) \(41623544884956481/2962701562500\) \(223443665096704101562500\) \([2, 2]\) \(56623104\) \(3.2274\)  
430950.hp4 430950hp4 \([1, 0, 0, -6084088, 4642079792]\) \(330240275458561/67652010000\) \(5102239542751406250000\) \([2, 2]\) \(28311552\) \(2.8808\)  
430950.hp5 430950hp2 \([1, 0, 0, -5746088, 5300841792]\) \(278202094583041/16646400\) \(1255453020900000000\) \([2, 2]\) \(14155776\) \(2.5342\) \(\Gamma_0(N)\)-optimal*
430950.hp6 430950hp1 \([1, 0, 0, -338088, 92937792]\) \(-56667352321/16711680\) \(-1260376366080000000\) \([2]\) \(7077888\) \(2.1876\) \(\Gamma_0(N)\)-optimal*
430950.hp7 430950hp5 \([1, 0, 0, 12928412, 27856342292]\) \(3168685387909439/6278181696900\) \(-473493498722378001562500\) \([2]\) \(56623104\) \(3.2274\)  
430950.hp8 430950hp7 \([1, 0, 0, 27673662, -264995434458]\) \(31077313442863199/420227050781250\) \(-31693057980537414550781250\) \([2]\) \(113246208\) \(3.5739\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 430950.hp1.

Rank

sage: E.rank()
 

The elliptic curves in class 430950.hp have rank \(1\).

Complex multiplication

The elliptic curves in class 430950.hp do not have complex multiplication.

Modular form 430950.2.a.hp

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

\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 2 & 4 & 8 & 16 & 8 & 4 \\ 16 & 1 & 8 & 4 & 2 & 4 & 8 & 16 \\ 2 & 8 & 1 & 2 & 4 & 8 & 4 & 2 \\ 4 & 4 & 2 & 1 & 2 & 4 & 2 & 4 \\ 8 & 2 & 4 & 2 & 1 & 2 & 4 & 8 \\ 16 & 4 & 8 & 4 & 2 & 1 & 8 & 16 \\ 8 & 8 & 4 & 2 & 4 & 8 & 1 & 8 \\ 4 & 16 & 2 & 4 & 8 & 16 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.