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Properties

Label 481390.h
Number of curves $4$
Conductor $481390$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("h1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 481390.h have rank \(0\).

Complex multiplication

The elliptic curves in class 481390.h do not have complex multiplication.

Modular form 481390.2.a.h

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - 2 q^{3} + q^{4} + q^{5} + 2 q^{6} - q^{7} - q^{8} + q^{9} - q^{10} - 2 q^{12} + q^{13} + q^{14} - 2 q^{15} + q^{16} - q^{18} - 2 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 481390.h

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
481390.h1 481390h4 \([1, 0, 1, -7759648, 8318970006]\) \(349046010201856969/7245875000\) \(1072649547207875000\) \([2]\) \(20528640\) \(2.5785\) \(\Gamma_0(N)\)-optimal*
481390.h2 481390h3 \([1, 0, 1, -501768, 120468758]\) \(94376601570889/12235496000\) \(1811292527715944000\) \([2]\) \(10264320\) \(2.2319\) \(\Gamma_0(N)\)-optimal*
481390.h3 481390h2 \([1, 0, 1, -160563, -5872312]\) \(3092354182009/1689383150\) \(250089336471870350\) \([2]\) \(6842880\) \(2.0292\) \(\Gamma_0(N)\)-optimal*
481390.h4 481390h1 \([1, 0, 1, -123533, -16699884]\) \(1408317602329/2153060\) \(318730151170340\) \([2]\) \(3421440\) \(1.6826\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 481390.h1.