Properties

 Label 2.5.af_p Base Field $\F_{5}$ Dimension $2$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

Invariants

 Base field: $\F_{5}$ Dimension: $2$ L-polynomial: $1 - 5 x + 15 x^{2} - 25 x^{3} + 25 x^{4}$ Frobenius angles: $\pm0.200000000000$, $\pm0.400000000000$ Angle rank: $0$ (numerical) Number field: $$\Q(\zeta_{5})$$ Galois group: $C_4$ Jacobians: 2

This isogeny class is simple but not geometrically simple.

Newton polygon

This isogeny class is supersingular.

 $p$-rank: $0$ Slopes: $[1/2, 1/2, 1/2, 1/2]$

Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

• $y^2=2x^5+3x+3$
• $y^2=2x^6+x^5+x^3+3x^2+x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 11 781 19151 406901 9759376 246109501 6152578751 152832422501 3808599606251 95245419909376

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 1 31 151 651 3126 15751 78751 391251 1950001 9753126

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\zeta_{5})$$.
Endomorphism algebra over $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{10}}$ is 1.9765625.ajgk 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $5$ and $\infty$.
All geometric endomorphisms are defined over $\F_{5^{10}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{5^{2}}$  The base change of $A$ to $\F_{5^{2}}$ is the simple isogeny class 2.25.f_z and its endomorphism algebra is $$\Q(\zeta_{5})$$.
• Endomorphism algebra over $\F_{5^{5}}$  The base change of $A$ to $\F_{5^{5}}$ is the simple isogeny class 2.3125.a_ajgk and its endomorphism algebra is the quaternion algebra over $$\Q(\sqrt{5})$$ ramified at both real infinite places.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.5.a_ak $5$ (not in LMFDB) 2.5.f_p $5$ (not in LMFDB) 2.5.a_f $15$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.5.a_ak $5$ (not in LMFDB) 2.5.f_p $5$ (not in LMFDB) 2.5.a_f $15$ (not in LMFDB) 2.5.a_k $20$ (not in LMFDB) 2.5.a_f $30$ (not in LMFDB) 2.5.a_a $40$ (not in LMFDB) 2.5.a_af $60$ (not in LMFDB)