Properties

Label 12544.h
Conductor $12544$
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Genus 2 curves in isogeny class 12544.h

Label Equation
12544.h.401408.1 \(y^2 = x^5 - x^4 + 4x^3 - x^2 + x\)

L-function data

Analytic rank:\(1\)
Mordell-Weil rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1\)
\(7\)\( ( 1 - T )( 1 + T )\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( ( 1 + 3 T^{2} )( 1 + 2 T + 3 T^{2} )\) 2.3.c_g
\(5\) \( ( 1 - 2 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )\) 2.5.c_c
\(11\) \( ( 1 - 4 T + 11 T^{2} )( 1 + 11 T^{2} )\) 2.11.ae_w
\(13\) \( ( 1 - 2 T + 13 T^{2} )( 1 + 13 T^{2} )\) 2.13.ac_ba
\(17\) \( ( 1 + 2 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )\) 2.17.i_bu
\(19\) \( ( 1 - 2 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )\) 2.19.g_w
\(23\) \( ( 1 + 23 T^{2} )( 1 + 8 T + 23 T^{2} )\) 2.23.i_bu
\(29\) \( ( 1 - 6 T + 29 T^{2} )( 1 - 2 T + 29 T^{2} )\) 2.29.ai_cs
$\cdots$$\cdots$$\cdots$
 
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Sato-Tate group

\(\mathrm{ST} =\) $\mathrm{SU}(2)\times\mathrm{SU}(2)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 112.a
  Elliptic curve isogeny class 112.b

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.