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Properties

Label 431970.y
Number of curves $6$
Conductor $431970$
CM no
Rank $2$
Graph

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Copy content sage:E = EllipticCurve("y1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 431970.y have rank \(2\).

Complex multiplication

The elliptic curves in class 431970.y do not have complex multiplication.

Modular form 431970.2.a.y

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - q^{10} - q^{12} - 6 q^{13} - q^{14} - q^{15} + q^{16} - q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 431970.y

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
431970.y1 431970y5 \([1, 1, 0, -210852182, 1178371824876]\) \(585196747116290735872321/836876053125000\) \(1482576977550178125000\) \([2]\) \(94371840\) \(3.3336\) \(\Gamma_0(N)\)-optimal*
431970.y2 431970y4 \([1, 1, 0, -30567022, -65049473804]\) \(1782900110862842086081/328139630024640\) \(581319371106081263040\) \([2]\) \(47185920\) \(2.9870\)  
431970.y3 431970y3 \([1, 1, 0, -13297902, 18056516724]\) \(146796951366228945601/5397929064360000\) \(9562760611186665960000\) \([2, 2]\) \(47185920\) \(2.9870\) \(\Gamma_0(N)\)-optimal*
431970.y4 431970y2 \([1, 1, 0, -2107822, -794292044]\) \(584614687782041281/184812061593600\) \(327405840648819609600\) \([2, 2]\) \(23592960\) \(2.6405\) \(\Gamma_0(N)\)-optimal*
431970.y5 431970y1 \([1, 1, 0, 370258, -84074316]\) \(3168685387909439/3563732336640\) \(-6313369222030295040\) \([2]\) \(11796480\) \(2.2939\) \(\Gamma_0(N)\)-optimal*
431970.y6 431970y6 \([1, 1, 0, 5215098, 64416771324]\) \(8854313460877886399/1016927675429790600\) \(-1801549409612075265126600\) \([2]\) \(94371840\) \(3.3336\)  
*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 431970.y1.