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Properties

Label 446160.ii
Number of curves $4$
Conductor $446160$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ii1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 446160.ii have rank \(1\).

Complex multiplication

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

Modular form 446160.2.a.ii

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 446160.ii

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
446160.ii1 446160ii4 \([0, 1, 0, -982787967360, -375006019159337292]\) \(5309860874757074224246393258249/4502770931800627200\) \(89022526499035765041581260800\) \([2]\) \(2724986880\) \(5.2898\)  
446160.ii2 446160ii2 \([0, 1, 0, -61437823360, -5856764844124492]\) \(1297212465095901089487274249/1193746061037404160000\) \(23601087410708036551169802240000\) \([2, 2]\) \(1362493440\) \(4.9433\)  
446160.ii3 446160ii3 \([0, 1, 0, -47399520640, -8604290914593100]\) \(-595697118196750093952139529/1272946549598037600000000\) \(-25166929355238417489995366400000000\) \([2]\) \(2724986880\) \(5.2898\)  
446160.ii4 446160ii1 \([0, 1, 0, -4730833280, -45863473850700]\) \(592265697637387401314569/296787655248366796800\) \(5867672885009260913030804275200\) \([2]\) \(681246720\) \(4.5967\) \(\Gamma_0(N)\)-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.ii1.