# Properties

 Label 2.19.aj_bu 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: $( 1 - 8 x + 19 x^{2} )( 1 - x + 19 x^{2} )$ Frobenius angles: $\pm0.130073469147$, $\pm0.463406802480$ Angle rank: $1$ (numerical) Jacobians: 10

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 10 curves, and hence is principally polarizable:

• $y^2=10x^6+18x^5+11x^4+5x^3+9x^2+15x+3$
• $y^2=6x^6+15x^5+9x^4+4x^3+18x^2+14x+12$
• $y^2=3x^6+16x^5+11x^4+18x^3+7x$
• $y^2=x^6+2x^3+15$
• $y^2=6x^6+13x^5+2x^4+16x^3+4x^2+6x+18$
• $y^2=12x^6+2x^5+5x^4+9x^3+16x^2+15x+17$
• $y^2=x^6+9x^5+2x^4+3x^3+11x^2+7x$
• $y^2=17x^6+11x^5+18x^4+9x^2+13x+2$
• $y^2=12x^6+10x^5+5x^4+10x^3+13x^2+8x+3$
• $y^2=15x^6+12x^5+12x^4+9x^2+4x+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 228 134064 47056464 16905470400 6130377927948 2214310804183296 799096559016987228 288443121410865657600 104127350297602681851984 37590003660735378639373104

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 11 373 6860 129721 2475821 47067046 893972279 16983663601 322687697780 6131071184053

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The isogeny class factors as 1.19.ai $\times$ 1.19.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 $\overline{\F}_{19}$
 The base change of $A$ to $\F_{19^{6}}$ is 1.47045881.pra 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
All geometric endomorphisms are defined over $\F_{19^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{19^{2}}$  The base change of $A$ to $\F_{19^{2}}$ is 1.361.aba $\times$ 1.361.bl. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{19^{3}}$  The base change of $A$ to $\F_{19^{3}}$ is 1.6859.ace $\times$ 1.6859.ce. The endomorphism algebra for each factor is:

## 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.ah_be $2$ (not in LMFDB) 2.19.h_be $2$ (not in LMFDB) 2.19.ap_dq $3$ (not in LMFDB) 2.19.ag_bf $3$ (not in LMFDB) 2.19.a_aba $3$ (not in LMFDB) 2.19.a_al $3$ (not in LMFDB) 2.19.a_bl $3$ (not in LMFDB) 2.19.g_bf $3$ (not in LMFDB) 2.19.j_bu $3$ (not in LMFDB) 2.19.p_dq $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.19.ah_be $2$ (not in LMFDB) 2.19.h_be $2$ (not in LMFDB) 2.19.ap_dq $3$ (not in LMFDB) 2.19.ag_bf $3$ (not in LMFDB) 2.19.a_aba $3$ (not in LMFDB) 2.19.a_al $3$ (not in LMFDB) 2.19.a_bl $3$ (not in LMFDB) 2.19.g_bf $3$ (not in LMFDB) 2.19.j_bu $3$ (not in LMFDB) 2.19.p_dq $3$ (not in LMFDB) 2.19.aq_dy $6$ (not in LMFDB) 2.19.ao_dj $6$ (not in LMFDB) 2.19.ai_bt $6$ (not in LMFDB) 2.19.ac_bn $6$ (not in LMFDB) 2.19.ab_as $6$ (not in LMFDB) 2.19.a_al $6$ (not in LMFDB) 2.19.b_as $6$ (not in LMFDB) 2.19.c_bn $6$ (not in LMFDB) 2.19.i_bt $6$ (not in LMFDB) 2.19.o_dj $6$ (not in LMFDB) 2.19.q_dy $6$ (not in LMFDB) 2.19.a_abl $12$ (not in LMFDB) 2.19.a_l $12$ (not in LMFDB) 2.19.a_ba $12$ (not in LMFDB)