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Properties

Label 446160fn
Number of curves $6$
Conductor $446160$
CM no
Rank $2$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("fn1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

Complex multiplication

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

Modular form 446160.2.a.fn

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} - q^{5} + q^{9} - q^{11} - q^{15} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 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

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 446160fn

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
446160.fn5 446160fn1 \([0, 1, 0, -29631, -3291300]\) \(-37256083456/38671875\) \(-2986588068750000\) \([2]\) \(2359296\) \(1.6647\) \(\Gamma_0(N)\)-optimal*
446160.fn4 446160fn2 \([0, 1, 0, -557756, -160461300]\) \(15529488955216/6125625\) \(7569208801440000\) \([2, 2]\) \(4718592\) \(2.0113\) \(\Gamma_0(N)\)-optimal*
446160.fn3 446160fn3 \([0, 1, 0, -642256, -108713500]\) \(5927735656804/2401490025\) \(11869730474066150400\) \([2, 2]\) \(9437184\) \(2.3579\) \(\Gamma_0(N)\)-optimal*
446160.fn1 446160fn4 \([0, 1, 0, -8923256, -10262639100]\) \(15897679904620804/2475\) \(12233064729600\) \([2]\) \(9437184\) \(2.3579\)  
446160.fn2 446160fn5 \([0, 1, 0, -4732056, 3884567220]\) \(1185450336504002/26043266205\) \(257445625257348802560\) \([2]\) \(18874368\) \(2.7044\) \(\Gamma_0(N)\)-optimal*
446160.fn6 446160fn6 \([0, 1, 0, 2095544, -788783020]\) \(102949393183198/86815346805\) \(-858196167263222261760\) \([2]\) \(18874368\) \(2.7044\)  
*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 446160fn1.