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Properties

Label 446160.di
Number of curves $4$
Conductor $446160$
CM no
Rank $2$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("di1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 446160.di have rank \(2\).

Complex multiplication

The elliptic curves in class 446160.di do not have complex multiplication.

Modular form 446160.2.a.di

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + q^{9} + q^{11} - q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 446160.di

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
446160.di1 446160di4 \([0, -1, 0, -71395120, -232170142400]\) \(2035678735521204409/141376950\) \(2795108493936844800\) \([2]\) \(24772608\) \(2.9936\)  
446160.di2 446160di3 \([0, -1, 0, -7634800, 2166847552]\) \(2489411558640889/1338278906250\) \(26458589877033600000000\) \([4]\) \(24772608\) \(2.9936\) \(\Gamma_0(N)\)-optimal*
446160.di3 446160di2 \([0, -1, 0, -4471120, -3611297600]\) \(499980107400409/4140922500\) \(81868562396375040000\) \([2, 2]\) \(12386304\) \(2.6471\) \(\Gamma_0(N)\)-optimal*
446160.di4 446160di1 \([0, -1, 0, -90640, -131444288]\) \(-4165509529/375289200\) \(-7419696284314828800\) \([2]\) \(6193152\) \(2.3005\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 446160.di1.