LMFDB - Abelian Variety Search Results

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Base field	Base char.	Dimension	$p$-rank	Initial coefficients

Slopes	Points on variety	Points on curve	Simple factors

Angle rank	# Jacobians	# Hyp. Jacobians	# twists	Max twist degree

End. degree	$p$-rank deficit

Simple	Geom. simple	Primitive	Princ polarizable	Jacobian

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Dim 1 factors	Dim 2 factors	Dim 3 factors	Dim 4 factors	Dim 5 factors

(distinct)	(distinct)	(distinct)	Number field	Galois group

Results (displaying matches 1-50 of at least 1000) Next

LMFDB label	dimension	base field	L-polynomial	$p$-rank	isogeny factors
4.2.ai_bg_adc_fg	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )^{4}$	$ 0$	1.2.ac⁴
4.2.ah_z_acg_ds	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )$	$ 2$	1.2.ac² $\times$ 2.2.ad_f
4.2.ah_ba_ack_ea	4	$\F_{2}$	$( 1 - x + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{3}$	$ 1$	1.2.ac³ $\times$ 1.2.ab
4.2.ag_s_abk_ce	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - 2 x + 2 x^{2} - 4 x^{3} + 4 x^{4} )$	$ 0$	1.2.ac² $\times$ 2.2.ac_c
4.2.ag_t_abq_cr	4	$\F_{2}$	$( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )^{2}$	$ 4$	2.2.ad_f²
4.2.ag_t_abp_co	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )( 1 - 4 x + 9 x^{2} - 15 x^{3} + 18 x^{4} - 16 x^{5} + 8 x^{6} )$	$ 3$	1.2.ac $\times$ 3.2.ae_j_ap
4.2.ag_t_abo_cm	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )$	$ 2$	1.2.ac² $\times$ 2.2.ac_d
4.2.ag_u_abt_cw	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )( 1 - x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )$	$ 3$	1.2.ac $\times$ 1.2.ab $\times$ 2.2.ad_f
4.2.ag_u_abs_cu	4	$\F_{2}$	$( 1 + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{3}$	$ 0$	1.2.ac³ $\times$ 1.2.a
4.2.ag_v_abw_dc	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - x + 2 x^{2} )^{2}$	$ 2$	1.2.ac² $\times$ 1.2.ab²
4.2.af_k_aj_g	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )( 1 - 3 x + 2 x^{2} + x^{3} + 4 x^{4} - 12 x^{5} + 8 x^{6} )$	$ 3$	1.2.ac $\times$ 3.2.ad_c_b
4.2.af_l_ao_q	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - x - x^{2} - 2 x^{3} + 4 x^{4} )$	$ 2$	1.2.ac² $\times$ 2.2.ab_ab
4.2.af_m_au_bd	4	$\F_{2}$	$1 - 5 x + 12 x^{2} - 20 x^{3} + 29 x^{4} - 40 x^{5} + 48 x^{6} - 40 x^{7} + 16 x^{8}$	$ 4$	simple
4.2.af_m_as_y	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - x - 2 x^{3} + 4 x^{4} )$	$ 1$	1.2.ac² $\times$ 2.2.ab_a
4.2.af_n_aba_bq	4	$\F_{2}$	$( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - 2 x + 2 x^{2} - 4 x^{3} + 4 x^{4} )$	$ 2$	2.2.ad_f $\times$ 2.2.ac_c
4.2.af_n_az_bn	4	$\F_{2}$	$1 - 5 x + 13 x^{2} - 25 x^{3} + 39 x^{4} - 50 x^{5} + 52 x^{6} - 40 x^{7} + 16 x^{8}$	$ 4$	simple
4.2.af_n_ax_bi	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 7 x^{3} + 10 x^{4} - 12 x^{5} + 8 x^{6} )$	$ 3$	1.2.ac $\times$ 3.2.ad_f_ah
4.2.af_n_aw_bg	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - x + x^{2} - 2 x^{3} + 4 x^{4} )$	$ 2$	1.2.ac² $\times$ 2.2.ab_b
4.2.af_o_abd_bv	4	$\F_{2}$	$( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )$	$ 4$	2.2.ad_f $\times$ 2.2.ac_d
4.2.af_o_abc_bs	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )( 1 - x + 2 x^{2} )( 1 - 2 x + 2 x^{2} - 4 x^{3} + 4 x^{4} )$	$ 1$	1.2.ac $\times$ 1.2.ab $\times$ 2.2.ac_c
4.2.af_o_abb_bq	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )( 1 - 3 x + 6 x^{2} - 9 x^{3} + 12 x^{4} - 12 x^{5} + 8 x^{6} )$	$ 3$	1.2.ac $\times$ 3.2.ad_g_aj
4.2.af_o_aba_bo	4	$\F_{2}$	$( 1 + x + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{3}$	$ 1$	1.2.ac³ $\times$ 1.2.b
4.2.af_p_abg_bz	4	$\F_{2}$	$( 1 - x + 2 x^{2} )( 1 - 4 x + 9 x^{2} - 15 x^{3} + 18 x^{4} - 16 x^{5} + 8 x^{6} )$	$ 4$	1.2.ab $\times$ 3.2.ae_j_ap
4.2.af_p_abg_ca	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )( 1 + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )$	$ 2$	1.2.ac $\times$ 1.2.a $\times$ 2.2.ad_f
4.2.af_p_abf_by	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )( 1 - x + 2 x^{2} )( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )$	$ 3$	1.2.ac $\times$ 1.2.ab $\times$ 2.2.ac_d
4.2.af_p_abe_bw	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - x + 3 x^{2} - 2 x^{3} + 4 x^{4} )$	$ 2$	1.2.ac² $\times$ 2.2.ab_d
4.2.af_q_abj_cf	4	$\F_{2}$	$( 1 - x + 2 x^{2} )^{2}( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )$	$ 4$	1.2.ab² $\times$ 2.2.ad_f
4.2.af_q_abi_ce	4	$\F_{2}$	$( 1 - x + 2 x^{2} )( 1 + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{2}$	$ 1$	1.2.ac² $\times$ 1.2.ab $\times$ 1.2.a
4.2.af_r_abl_ck	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )( 1 - x + 2 x^{2} )^{3}$	$ 3$	1.2.ac $\times$ 1.2.ab³
4.2.ae_e_h_av	4	$\F_{2}$	$1 - 4 x + 4 x^{2} + 7 x^{3} - 21 x^{4} + 14 x^{5} + 16 x^{6} - 32 x^{7} + 16 x^{8}$	$ 4$	simple
4.2.ae_e_i_ay	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - 2 x^{2} )^{2}$	$ 0$	1.2.ac² $\times$ 2.2.a_ae
4.2.ae_f_c_al	4	$\F_{2}$	$1 - 4 x + 5 x^{2} + 2 x^{3} - 11 x^{4} + 4 x^{5} + 20 x^{6} - 32 x^{7} + 16 x^{8}$	$ 4$	simple
4.2.ae_f_e_aq	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - 3 x^{2} + 4 x^{4} )$	$ 2$	1.2.ac² $\times$ 2.2.a_ad
4.2.ae_g_ae_c	4	$\F_{2}$	$1 - 4 x + 6 x^{2} - 4 x^{3} + 2 x^{4} - 8 x^{5} + 24 x^{6} - 32 x^{7} + 16 x^{8}$	$ 0$	simple
4.2.ae_g_ab_ag	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )( 1 - 2 x + 3 x^{3} - 8 x^{5} + 8 x^{6} )$	$ 3$	1.2.ac $\times$ 3.2.ac_a_d
4.2.ae_g_a_ai	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - 2 x^{2} + 4 x^{4} )$	$ 0$	1.2.ac² $\times$ 2.2.a_ac
4.2.ae_h_ak_p	4	$\F_{2}$	$( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - x - x^{2} - 2 x^{3} + 4 x^{4} )$	$ 4$	2.2.ad_f $\times$ 2.2.ab_ab
4.2.ae_h_ah_h	4	$\F_{2}$	$( 1 - x + 2 x^{2} )( 1 - 3 x + 2 x^{2} + x^{3} + 4 x^{4} - 12 x^{5} + 8 x^{6} )$	$ 4$	1.2.ab $\times$ 3.2.ad_c_b
4.2.ae_h_ag_e	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )( 1 - 2 x + x^{2} + 2 x^{4} - 8 x^{5} + 8 x^{6} )$	$ 2$	1.2.ac $\times$ 3.2.ac_b_a
4.2.ae_h_af_c	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )( 1 - 2 x + x^{2} + x^{3} + 2 x^{4} - 8 x^{5} + 8 x^{6} )$	$ 3$	1.2.ac $\times$ 3.2.ac_b_b
4.2.ae_h_ae_a	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - x^{2} + 4 x^{4} )$	$ 2$	1.2.ac² $\times$ 2.2.a_ab
4.2.ae_i_aq_bc	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} - 4 x^{3} + 4 x^{4} )^{2}$	$ 0$	2.2.ac_c²
4.2.ae_i_an_u	4	$\F_{2}$	$( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - x - 2 x^{3} + 4 x^{4} )$	$ 3$	2.2.ad_f $\times$ 2.2.ab_a
4.2.ae_i_am_r	4	$\F_{2}$	$1 - 4 x + 8 x^{2} - 12 x^{3} + 17 x^{4} - 24 x^{5} + 32 x^{6} - 32 x^{7} + 16 x^{8}$	$ 4$	simple
4.2.ae_i_al_o	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )( 1 - x + 2 x^{2} )( 1 - x - x^{2} - 2 x^{3} + 4 x^{4} )$	$ 3$	1.2.ac $\times$ 1.2.ab $\times$ 2.2.ab_ab
4.2.ae_i_ak_m	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )( 1 - 2 x + 2 x^{2} - 2 x^{3} + 4 x^{4} - 8 x^{5} + 8 x^{6} )$	$ 0$	1.2.ac $\times$ 3.2.ac_c_ac
4.2.ae_i_aj_k	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} )( 1 - 2 x + 2 x^{2} - x^{3} + 4 x^{4} - 8 x^{5} + 8 x^{6} )$	$ 3$	1.2.ac $\times$ 3.2.ac_c_ab
4.2.ae_i_ai_i	4	$\F_{2}$	$( 1 + 2 x + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{3}$	$ 0$	1.2.ac³ $\times$ 1.2.c
4.2.ae_j_as_be	4	$\F_{2}$	$( 1 - 2 x + 2 x^{2} - 4 x^{3} + 4 x^{4} )( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )$	$ 2$	2.2.ac_c $\times$ 2.2.ac_d
4.2.ae_j_ar_bb	4	$\F_{2}$	$1 - 4 x + 9 x^{2} - 17 x^{3} + 27 x^{4} - 34 x^{5} + 36 x^{6} - 32 x^{7} + 16 x^{8}$	$ 4$	simple

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

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