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Properties

Label 446160ge
Number of curves $2$
Conductor $446160$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ge1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 446160ge have rank \(0\).

Complex multiplication

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

Modular form 446160.2.a.ge

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} - q^{5} + 2 q^{7} + q^{9} + q^{11} - q^{15} + 4 q^{17} - 2 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 446160ge

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
446160.ge2 446160ge1 \([0, 1, 0, 200873344, 343462232244]\) \(45338857965533777399/28814396538470400\) \(-569678186665811038017945600\) \([2]\) \(151732224\) \(3.8229\) \(\Gamma_0(N)\)-optimal*
446160.ge1 446160ge2 \([0, 1, 0, -846115456, 2813937004724]\) \(3388383326345613179401/1787816842064922240\) \(35346228935190713353552527360\) \([2]\) \(303464448\) \(4.1695\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 446160ge1.