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Properties

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

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 1, 3847, -282999]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, 3847, -282999]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, 3847, -282999]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

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

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1 + T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 - 2 T + 11 T^{2}\) 1.11.ac
\(13\) \( 1 + 6 T + 13 T^{2}\) 1.13.g
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 26010.2.a.bm

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q + q^{2} + q^{4} - q^{5} + 4 q^{7} + q^{8} - q^{10} + 2 q^{13} + 4 q^{14} + q^{16} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 26010bm

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
26010.bl8 26010bm1 \([1, -1, 1, 3847, -282999]\) \(357911/2160\) \(-38007981650160\) \([2]\) \(73728\) \(1.2876\) \(\Gamma_0(N)\)-optimal
26010.bl6 26010bm2 \([1, -1, 1, -48173, -3674703]\) \(702595369/72900\) \(1282769380692900\) \([2, 2]\) \(147456\) \(1.6342\)  
26010.bl7 26010bm3 \([1, -1, 1, -35168, 8315907]\) \(-273359449/1536000\) \(-27027898062336000\) \([2]\) \(221184\) \(1.8369\)  
26010.bl5 26010bm4 \([1, -1, 1, -178223, 24988317]\) \(35578826569/5314410\) \(93513887852512410\) \([2]\) \(294912\) \(1.9808\)  
26010.bl4 26010bm5 \([1, -1, 1, -750443, -250031019]\) \(2656166199049/33750\) \(593874713283750\) \([2]\) \(294912\) \(1.9808\)  
26010.bl3 26010bm6 \([1, -1, 1, -867488, 310614531]\) \(4102915888729/9000000\) \(158366590209000000\) \([2, 2]\) \(442368\) \(2.1835\)  
26010.bl1 26010bm7 \([1, -1, 1, -13872488, 19890942531]\) \(16778985534208729/81000\) \(1425299311881000\) \([2]\) \(884736\) \(2.5301\)  
26010.bl2 26010bm8 \([1, -1, 1, -1179608, 67410627]\) \(10316097499609/5859375000\) \(103103248833984375000\) \([2]\) \(884736\) \(2.5301\)