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Properties

Label 401115h
Number of curves $4$
Conductor $401115$
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 401115h have rank \(0\).

Complex multiplication

The elliptic curves in class 401115h do not have complex multiplication.

Modular form 401115.2.a.h

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} - 4 q^{7} + 3 q^{8} + q^{9} + q^{10} + q^{12} - q^{13} + 4 q^{14} + q^{15} - q^{16} - q^{17} - q^{18} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 401115h

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
401115.h4 401115h1 \([1, 1, 1, 27459919, -87507568906]\) \(1292603583867446566871/2615843353271484375\) \(-4634126066764984130859375\) \([2]\) \(76677120\) \(3.4174\) \(\Gamma_0(N)\)-optimal*
401115.h3 401115h2 \([1, 1, 1, -208868206, -937721631406]\) \(568832774079017834683129/114800389711906640625\) \(203375893198415040172265625\) \([2, 2]\) \(153354240\) \(3.7640\) \(\Gamma_0(N)\)-optimal*
401115.h2 401115h3 \([1, 1, 1, -1039608831, 12065362927344]\) \(70141892778055497175333129/5090453819946781723125\) \(9018049459718740576201048125\) \([2]\) \(306708480\) \(4.1105\) \(\Gamma_0(N)\)-optimal*
401115.h1 401115h4 \([1, 1, 1, -3159377581, -68349779627656]\) \(1968666709544018637994033129/113621848881699526875\) \(201288036226712495530201875\) \([2]\) \(306708480\) \(4.1105\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 401115h1.