Properties

 Label 2.4.ac_e Base Field $\F_{2^{2}}$ Dimension $2$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

Invariants

 Base field: $\F_{2^{2}}$ Dimension: $2$ L-polynomial: $1 - 2 x + 4 x^{2} - 8 x^{3} + 16 x^{4}$ Frobenius angles: $\pm0.200000000000$, $\pm0.600000000000$ 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:

• $y^2+y=(a+1)x^5+(a+1)x^3+a$
• $y^2+y=ax^5+ax^3+a+1$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 11 341 3641 69905 1185921 17043521 266354561 4311810305 68585520641 1095222947841

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 21 57 273 1153 4161 16257 65793 261633 1044481

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\zeta_{5})$$.
Endomorphism algebra over $\overline{\F}_{2^{2}}$
 The base change of $A$ to $\F_{2^{10}}$ is 1.1024.cm 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{10}}$.

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 2.4.c_e $2$ 2.16.e_q 2.4.i_y $5$ 2.1024.ey_jci 2.4.ai_y $10$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.c_e $2$ 2.16.e_q 2.4.i_y $5$ 2.1024.ey_jci 2.4.ai_y $10$ (not in LMFDB) 2.4.a_ai $10$ (not in LMFDB) 2.4.ae_m $15$ (not in LMFDB) 2.4.c_a $15$ (not in LMFDB) 2.4.ae_i $20$ (not in LMFDB) 2.4.a_i $20$ (not in LMFDB) 2.4.e_i $20$ (not in LMFDB) 2.4.ag_q $30$ (not in LMFDB) 2.4.ac_a $30$ (not in LMFDB) 2.4.a_e $30$ (not in LMFDB) 2.4.e_m $30$ (not in LMFDB) 2.4.g_q $30$ (not in LMFDB) 2.4.a_a $40$ (not in LMFDB) 2.4.ac_i $60$ (not in LMFDB) 2.4.a_ae $60$ (not in LMFDB) 2.4.c_i $60$ (not in LMFDB)