Properties

Label 26010r
Number of curves $8$
Conductor $26010$
CM no
Rank $1$
Graph

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

Elliptic curves in class 26010r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
26010.u6 26010r1 \([1, -1, 0, -208134, -44852940]\) \(-56667352321/16711680\) \(-294063530918215680\) \([2]\) \(294912\) \(2.0664\) \(\Gamma_0(N)\)-optimal
26010.u5 26010r2 \([1, -1, 0, -3537414, -2559791052]\) \(278202094583041/16646400\) \(292914845250566400\) \([2, 2]\) \(589824\) \(2.4129\)  
26010.u4 26010r3 \([1, -1, 0, -3745494, -2241553500]\) \(330240275458561/67652010000\) \(1190424238276130010000\) \([2, 2]\) \(1179648\) \(2.7595\)  
26010.u2 26010r4 \([1, -1, 0, -56597814, -163874019132]\) \(1139466686381936641/4080\) \(71792854228080\) \([2]\) \(1179648\) \(2.7595\)  
26010.u7 26010r5 \([1, -1, 0, 7959006, -13456805400]\) \(3168685387909439/6278181696900\) \(-110472692005622949516900\) \([2]\) \(2359296\) \(3.1061\)  
26010.u3 26010r6 \([1, -1, 0, -18779274, 29338404768]\) \(41623544884956481/2962701562500\) \(52132549362222389062500\) \([2, 2]\) \(2359296\) \(3.1061\)  
26010.u8 26010r7 \([1, -1, 0, 17036496, 127996524810]\) \(31077313442863199/420227050781250\) \(-7394436127312316894531250\) \([2]\) \(4718592\) \(3.4527\)  
26010.u1 26010r8 \([1, -1, 0, -295135524, 1951617208518]\) \(161572377633716256481/914742821250\) \(16096077946613698571250\) \([2]\) \(4718592\) \(3.4527\)  

Rank

sage: E.rank()
 

The elliptic curves in class 26010r have rank \(1\).

Complex multiplication

The elliptic curves in class 26010r do not have complex multiplication.

Modular form 26010.2.a.r

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} + 4 q^{11} - 2 q^{13} + q^{16} + 4 q^{19} + O(q^{20})\) Copy content 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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.