# Properties

 Label 3.2.ad_d_ac Base field $\F_{2}$ Dimension $3$ $p$-rank $2$ Ordinary no Supersingular no Simple no Geometrically simple no Primitive yes Principally polarizable yes Contains a Jacobian yes

## Invariants

 Base field: $\F_{2}$ Dimension: $3$ L-polynomial: $( 1 - 2 x + 2 x^{2} )( 1 - x - x^{2} - 2 x^{3} + 4 x^{4} )$ $1 - 3x + 3x^{2} - 2x^{3} + 6x^{4} - 12x^{5} + 8x^{6}$ Frobenius angles: $\pm0.0516399385854$, $\pm0.250000000000$, $\pm0.718306605252$ Angle rank: $1$ (numerical) Jacobians: 1

This isogeny class is not simple.

## Newton polygon

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

## Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

• $x^4+x^3y+x^3z+x^2yz+y^4+y^3z+z^4=0$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $35$ $208$ $6475$ $30791$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $0$ $2$ $3$ $26$ $30$ $47$ $126$ $194$ $471$ $1082$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac $\times$ 2.2.ab_ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{12}}$ is 1.4096.bv 2 $\times$ 1.4096.ey. The endomorphism algebra for each factor is: 1.4096.bv 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$ 1.4096.ey : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{12}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{2}}$  The base change of $A$ to $\F_{2^{2}}$ is 1.4.a $\times$ 2.4.ad_f. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is 1.8.af 2 $\times$ 1.8.e. The endomorphism algebra for each factor is: 1.8.af 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$ 1.8.e : $$\Q(\sqrt{-1})$$.
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i $\times$ 2.16.b_ap. The endomorphism algebra for each factor is: 1.16.i : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.b_ap : $$\Q(\sqrt{-3}, \sqrt{-7})$$.
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj 2 $\times$ 1.64.a. The endomorphism algebra for each factor is: 1.64.aj 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$ 1.64.a : $$\Q(\sqrt{-1})$$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
TwistExtension degreeCommon base change
3.2.ab_ab_g$2$3.4.ad_j_ay
3.2.b_ab_ag$2$3.4.ad_j_ay
3.2.d_d_c$2$3.4.ad_j_ay
3.2.a_d_ac$3$3.8.ag_j_e
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
3.2.ab_ab_g$2$3.4.ad_j_ay
3.2.b_ab_ag$2$3.4.ad_j_ay
3.2.d_d_c$2$3.4.ad_j_ay
3.2.a_d_ac$3$3.8.ag_j_e
3.2.ae_l_as$6$(not in LMFDB)
3.2.ac_f_ag$6$(not in LMFDB)
3.2.a_d_c$6$(not in LMFDB)
3.2.c_f_g$6$(not in LMFDB)
3.2.e_l_s$6$(not in LMFDB)
3.2.ab_b_ae$8$(not in LMFDB)
3.2.b_b_e$8$(not in LMFDB)
3.2.ac_ab_g$12$(not in LMFDB)
3.2.c_ab_ag$12$(not in LMFDB)
3.2.e_l_s$12$(not in LMFDB)
3.2.ac_h_ai$24$(not in LMFDB)
3.2.a_ab_a$24$(not in LMFDB)
3.2.a_f_a$24$(not in LMFDB)
3.2.c_h_i$24$(not in LMFDB)