Properties

Label 348726.r
Number of curves $6$
Conductor $348726$
CM no
Rank $2$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("r1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 348726.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
348726.r1 348726r5 \([1, 0, 1, -28567382, 58766768156]\) \(54804145548726848737/637608031452\) \(29996831572335049212\) \([2]\) \(28311552\) \(2.8882\)  
348726.r2 348726r4 \([1, 0, 1, -6394762, -6224581684]\) \(614716917569296417/19093020912\) \(898247989756463472\) \([2]\) \(14155776\) \(2.5416\)  
348726.r3 348726r3 \([1, 0, 1, -1831722, 868022860]\) \(14447092394873377/1439452851984\) \(67720327579549877904\) \([2, 2]\) \(14155776\) \(2.5416\)  
348726.r4 348726r2 \([1, 0, 1, -416602, -88598260]\) \(169967019783457/26337394944\) \(1239065948385425664\) \([2, 2]\) \(7077888\) \(2.1950\)  
348726.r5 348726r1 \([1, 0, 1, 45478, -7641844]\) \(221115865823/664731648\) \(-31272886008741888\) \([2]\) \(3538944\) \(1.8485\) \(\Gamma_0(N)\)-optimal
348726.r6 348726r6 \([1, 0, 1, 2262018, 4198689724]\) \(27207619911317663/177609314617308\) \(-8355786679977432708348\) \([2]\) \(28311552\) \(2.8882\)  

Rank

sage: E.rank()
 

The elliptic curves in class 348726.r have rank \(2\).

Complex multiplication

The elliptic curves in class 348726.r do not have complex multiplication.

Modular form 348726.2.a.r

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - 2q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + 2q^{10} - 4q^{11} + q^{12} + 2q^{13} - q^{14} - 2q^{15} + q^{16} - 6q^{17} - q^{18} + O(q^{20})\)  Toggle raw display

Isogeny matrix

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

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

The vertices are labelled with LMFDB labels.