Properties

 Label 2.19.al_ck Base Field $\F_{19}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

Invariants

 Base field: $\F_{19}$ Dimension: $2$ L-polynomial: $( 1 - 8 x + 19 x^{2} )( 1 - 3 x + 19 x^{2} )$ Frobenius angles: $\pm0.130073469147$, $\pm0.388176076177$ Angle rank: $2$ (numerical) Jacobians: 4

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class contains the Jacobians of 4 curves, and hence is principally polarizable:

• $y^2=5x^6+4x^5+2x^4+8x^3+10x^2+12x+9$
• $y^2=8x^6+14x^5+12x^4+4x^3+9x^2+2x+10$
• $y^2=11x^6+15x^5+6x^4+18x^3+14x+10$
• $y^2=10x^6+18x^4+2x^3+4x^2+x+15$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 204 131376 47655216 16974304704 6126953978724 2213482610416896 799091751465814164 288449990483531192064 104127672854620178665776 37589962955782452689708976

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 9 365 6948 130249 2474439 47049446 893966901 16984068049 322688697372 6131064544925

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The isogeny class factors as 1.19.ai $\times$ 1.19.ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{19}$.

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 2.19.af_o $2$ (not in LMFDB) 2.19.f_o $2$ (not in LMFDB) 2.19.l_ck $2$ (not in LMFDB) 2.19.ac_bj $3$ (not in LMFDB) 2.19.e_r $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.19.af_o $2$ (not in LMFDB) 2.19.f_o $2$ (not in LMFDB) 2.19.l_ck $2$ (not in LMFDB) 2.19.ac_bj $3$ (not in LMFDB) 2.19.e_r $3$ (not in LMFDB) 2.19.ak_ch $6$ (not in LMFDB) 2.19.ae_r $6$ (not in LMFDB) 2.19.ae_bp $6$ (not in LMFDB) 2.19.c_bj $6$ (not in LMFDB) 2.19.e_bp $6$ (not in LMFDB) 2.19.k_ch $6$ (not in LMFDB)