Number field 26.2.23347285006439981249999555372409414171948497.1

• Label: 26.2.233...497.1
• Degree: $26$
• Signature: $[2, 12]$
• Discriminant: $2.335\times 10^{43}$
• Root discriminant: $46.56$
• Ramified primes: $17, 191$
• Class number: not computed
• Class group: not computed
• Galois group: $D_{26}$ (as 26T3)

Normalized defining polynomial

\( x^{26} - x^{25} - 50 x^{24} + 48 x^{23} + 1188 x^{22} - 1066 x^{21} - 17960 x^{20} + 14231 x^{19} + 194023 x^{18} - 126696 x^{17} - 1579415 x^{16} + 794998 x^{15} + 9883811 x^{14} - 3670079 x^{13} - 47524550 x^{12} + 13070104 x^{11} + 173418938 x^{10} - 37145018 x^{9} - 469905596 x^{8} + 84248201 x^{7} + 912345834 x^{6} - 152992905 x^{5} - 1188256605 x^{4} + 203114817 x^{3} + 913828487 x^{2} - 132782216 x - 329129521 \)

Invariants

Degree:		$26$
Signature:		$[2, 12]$
Discriminant:		\(23347285006439981249999555372409414171948497\)\(\medspace = 17^{13}\cdot 191^{12}\)
Root discriminant:		$46.56$
Ramified primes:		$17, 191$
$|\Aut(K/\Q)|$:		$2$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} + \frac{3}{7} a^{15} - \frac{2}{7} a^{14} + \frac{1}{7} a^{13} - \frac{1}{7} a^{12} + \frac{2}{7} a^{11} + \frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{3}{7} a^{8} - \frac{3}{7} a^{7} - \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{7} a^{17} + \frac{3}{7} a^{15} + \frac{3}{7} a^{13} - \frac{2}{7} a^{12} + \frac{2}{7} a^{11} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} + \frac{3}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{7} a^{18} - \frac{2}{7} a^{15} + \frac{2}{7} a^{14} + \frac{2}{7} a^{13} - \frac{2}{7} a^{12} + \frac{1}{7} a^{11} - \frac{1}{7} a^{10} - \frac{3}{7} a^{9} + \frac{2}{7} a^{8} + \frac{2}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{7} a^{19} + \frac{1}{7} a^{15} - \frac{2}{7} a^{14} - \frac{1}{7} a^{12} + \frac{3}{7} a^{11} - \frac{1}{7} a^{10} + \frac{1}{7} a^{9} + \frac{3}{7} a^{8} + \frac{3}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{7} a^{20} + \frac{2}{7} a^{15} + \frac{2}{7} a^{14} - \frac{2}{7} a^{13} - \frac{3}{7} a^{12} - \frac{3}{7} a^{11} - \frac{1}{7} a^{8} + \frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{2}{7} a$, $\frac{1}{7} a^{21} + \frac{3}{7} a^{15} + \frac{2}{7} a^{14} + \frac{2}{7} a^{13} - \frac{1}{7} a^{12} + \frac{3}{7} a^{11} - \frac{2}{7} a^{10} - \frac{1}{7} a^{8} + \frac{2}{7} a^{6} - \frac{3}{7} a^{5} + \frac{3}{7} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{7} a^{22} + \frac{1}{7} a^{14} + \frac{3}{7} a^{13} - \frac{1}{7} a^{12} - \frac{1}{7} a^{11} - \frac{3}{7} a^{10} - \frac{3}{7} a^{9} + \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{163009} a^{23} - \frac{10869}{163009} a^{22} - \frac{4679}{163009} a^{21} + \frac{953}{163009} a^{20} + \frac{5333}{163009} a^{19} - \frac{4801}{163009} a^{18} - \frac{2903}{163009} a^{17} + \frac{58}{5621} a^{16} + \frac{73286}{163009} a^{15} - \frac{49132}{163009} a^{14} - \frac{9078}{23287} a^{13} - \frac{55954}{163009} a^{12} + \frac{21843}{163009} a^{11} - \frac{29387}{163009} a^{10} - \frac{16091}{163009} a^{9} - \frac{632}{2233} a^{8} - \frac{66048}{163009} a^{7} + \frac{72025}{163009} a^{6} - \frac{69709}{163009} a^{5} - \frac{3408}{14819} a^{4} + \frac{5512}{23287} a^{3} - \frac{59288}{163009} a^{2} - \frac{56006}{163009} a - \frac{415}{23287}$, $\frac{1}{35372953} a^{24} + \frac{38}{35372953} a^{23} + \frac{1538197}{35372953} a^{22} + \frac{1502572}{35372953} a^{21} + \frac{2342402}{35372953} a^{20} - \frac{51143}{5053279} a^{19} - \frac{69698}{1141063} a^{18} + \frac{2477403}{35372953} a^{17} + \frac{136230}{5053279} a^{16} - \frac{15787017}{35372953} a^{15} - \frac{3931468}{35372953} a^{14} - \frac{14918301}{35372953} a^{13} + \frac{10073958}{35372953} a^{12} + \frac{4131290}{35372953} a^{11} + \frac{16748808}{35372953} a^{10} - \frac{16244306}{35372953} a^{9} - \frac{14803288}{35372953} a^{8} - \frac{16557084}{35372953} a^{7} - \frac{14611067}{35372953} a^{6} + \frac{98093}{1141063} a^{5} - \frac{16829674}{35372953} a^{4} + \frac{167570}{3215723} a^{3} + \frac{6606563}{35372953} a^{2} - \frac{18653}{163009} a + \frac{118515}{721897}$, $\frac{1}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{25} + \frac{81228747778157298766336125167645830967297969991460564105395675811544560415839}{9513579977039580566533403611729742900338896456760823921665974878282150159596865274131} a^{24} - \frac{122368490765312381810536387242653720646725423817474979500398642029580849520479847}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{23} + \frac{4258567952574708601032643676373139787976877574837961664066089622546960003671410016943}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{22} + \frac{133991884208597129485408153953326578500301921980547365500951528324210444646760088718}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{21} - \frac{539379108677981958709230079183218364216350496430988941796573793982081648165040870316}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{20} - \frac{2019088890995769931183176267616816766573601739331016992789203853547112317488501711037}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{19} - \frac{4750151328136286053308318296780649425448370159327222172012952865287029508054081456183}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{18} - \frac{11176550909253766280286208510625481109100210805535944581102825687463361487056060948}{6054096349025187633248529571100745482033843199756887950151074922543186465198005174447} a^{17} + \frac{2310997361101964822042733071906076648178012479154443449343679031582095778492928650509}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{16} + \frac{2801801609959891066067397895921751203920895493474669788890236128285402021378864496248}{6054096349025187633248529571100745482033843199756887950151074922543186465198005174447} a^{15} - \frac{3884265680689510516752661729066412997568935232838794364561538710574845902806833294355}{9513579977039580566533403611729742900338896456760823921665974878282150159596865274131} a^{14} - \frac{3739648237417283950085228839081235786625139506839740887839386714545974302287930384889}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{13} - \frac{10928599450799558569359338724606365786259043952603786440614500522105959847639866110934}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{12} + \frac{3675551723257075228708312390154677540122533288607951954794529452169451408453899003412}{9513579977039580566533403611729742900338896456760823921665974878282150159596865274131} a^{11} + \frac{198217919235856785958340195918440430319586584204434358250413296155672648445228342000}{937958589285592450221603172987439440878482749258109400727631326027817621368705027027} a^{10} - \frac{277497295833235444848526188307349360423444996598290980237063673696175187004588953045}{1359082853862797223790486230247106128619842350965831988809424982611735737085266467733} a^{9} + \frac{16559058916476016616097480042401342911601279922429613135320739707883590668795856916143}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{8} - \frac{18725822961939545087693072269645118874170167971208377886118769072982261771409893024577}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{7} + \frac{3020929572741896167888155759062830875246528815209141447611106869301751044962441289830}{6054096349025187633248529571100745482033843199756887950151074922543186465198005174447} a^{6} - \frac{10691686472543941638069758095223232220773662737152535820278355222030908417160636791570}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{5} + \frac{16379407463217327245158368551897817367429952695051682307182221093710725058205970454505}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{4} + \frac{27667445491370508326528132688920981533472601018661052450448117542603234190429503388462}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{3} + \frac{18262282478176953919572524801020400738861855065453546378104510172180940705516440309514}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{2} + \frac{3173482666396580323725343425319677144203225402024543759060407373134554094002711963093}{9513579977039580566533403611729742900338896456760823921665974878282150159596865274131} a - \frac{575696147268037436371477533602341883186955364661130850108220325310487558382430734411}{1359082853862797223790486230247106128619842350965831988809424982611735737085266467733}$

Class group and class number

not computed

Unit group

Rank:		$13$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		not computed
Regulator:		not computed

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$D_{26}$ (as 26T3):

A solvable group of order 52

The 16 conjugacy class representatives for $D_{26}$

Character table for $D_{26}$

Intermediate fields

\(\Q(\sqrt{17}) \), 13.1.48551226272641.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 26 sibling:

26.0.4459331436230036418749915076130198106842162927.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/LocalNumberField/2.13.0.1}{13} }^{2}$	$26$	$26$	${\href{/LocalNumberField/7.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/11.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$	R	${\href{/LocalNumberField/19.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$	$26$	${\href{/LocalNumberField/29.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
17	Data not computed
$191$	$\Q_{191}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{191}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	191.2.1.2	$x^{2} + 382$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	191.2.1.2	$x^{2} + 382$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	191.2.1.2	$x^{2} + 382$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	191.2.1.2	$x^{2} + 382$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	191.2.1.2	$x^{2} + 382$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	191.2.1.2	$x^{2} + 382$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	191.2.1.2	$x^{2} + 382$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	191.2.1.2	$x^{2} + 382$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	191.2.1.2	$x^{2} + 382$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	191.2.1.2	$x^{2} + 382$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	191.2.1.2	$x^{2} + 382$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	191.2.1.2	$x^{2} + 382$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.3247.2t1.a.a	$1$	$ 17 \cdot 191 $	$x^{2} - x + 812$	$C_2$ (as 2T1)	$1$	$-1$
	1.191.2t1.a.a	$1$	$ 191 $	$x^{2} - x + 48$	$C_2$ (as 2T1)	$1$	$-1$
*	1.17.2t1.a.a	$1$	$ 17 $	$x^{2} - x - 4$	$C_2$ (as 2T1)	$1$	$1$
*	2.55199.26t3.b.f	$2$	$ 17^{2} \cdot 191 $	$x^{26} - x^{25} - 50 x^{24} + 48 x^{23} + 1188 x^{22} - 1066 x^{21} - 17960 x^{20} + 14231 x^{19} + 194023 x^{18} - 126696 x^{17} - 1579415 x^{16} + 794998 x^{15} + 9883811 x^{14} - 3670079 x^{13} - 47524550 x^{12} + 13070104 x^{11} + 173418938 x^{10} - 37145018 x^{9} - 469905596 x^{8} + 84248201 x^{7} + 912345834 x^{6} - 152992905 x^{5} - 1188256605 x^{4} + 203114817 x^{3} + 913828487 x^{2} - 132782216 x - 329129521$	$D_{26}$ (as 26T3)	$1$	$0$
*	2.191.13t2.a.b	$2$	$ 191 $	$x^{13} - 2 x^{12} + 4 x^{10} - 5 x^{9} + x^{8} + 5 x^{7} - 11 x^{6} + 19 x^{5} - 22 x^{4} + 16 x^{3} - 10 x^{2} + 6 x - 1$	$D_{13}$ (as 13T2)	$1$	$0$
*	2.191.13t2.a.a	$2$	$ 191 $	$x^{13} - 2 x^{12} + 4 x^{10} - 5 x^{9} + x^{8} + 5 x^{7} - 11 x^{6} + 19 x^{5} - 22 x^{4} + 16 x^{3} - 10 x^{2} + 6 x - 1$	$D_{13}$ (as 13T2)	$1$	$0$
*	2.55199.26t3.b.a	$2$	$ 17^{2} \cdot 191 $	$x^{26} - x^{25} - 50 x^{24} + 48 x^{23} + 1188 x^{22} - 1066 x^{21} - 17960 x^{20} + 14231 x^{19} + 194023 x^{18} - 126696 x^{17} - 1579415 x^{16} + 794998 x^{15} + 9883811 x^{14} - 3670079 x^{13} - 47524550 x^{12} + 13070104 x^{11} + 173418938 x^{10} - 37145018 x^{9} - 469905596 x^{8} + 84248201 x^{7} + 912345834 x^{6} - 152992905 x^{5} - 1188256605 x^{4} + 203114817 x^{3} + 913828487 x^{2} - 132782216 x - 329129521$	$D_{26}$ (as 26T3)	$1$	$0$
*	2.55199.26t3.b.e	$2$	$ 17^{2} \cdot 191 $	$x^{26} - x^{25} - 50 x^{24} + 48 x^{23} + 1188 x^{22} - 1066 x^{21} - 17960 x^{20} + 14231 x^{19} + 194023 x^{18} - 126696 x^{17} - 1579415 x^{16} + 794998 x^{15} + 9883811 x^{14} - 3670079 x^{13} - 47524550 x^{12} + 13070104 x^{11} + 173418938 x^{10} - 37145018 x^{9} - 469905596 x^{8} + 84248201 x^{7} + 912345834 x^{6} - 152992905 x^{5} - 1188256605 x^{4} + 203114817 x^{3} + 913828487 x^{2} - 132782216 x - 329129521$	$D_{26}$ (as 26T3)	$1$	$0$
*	2.55199.26t3.b.c	$2$	$ 17^{2} \cdot 191 $	$x^{26} - x^{25} - 50 x^{24} + 48 x^{23} + 1188 x^{22} - 1066 x^{21} - 17960 x^{20} + 14231 x^{19} + 194023 x^{18} - 126696 x^{17} - 1579415 x^{16} + 794998 x^{15} + 9883811 x^{14} - 3670079 x^{13} - 47524550 x^{12} + 13070104 x^{11} + 173418938 x^{10} - 37145018 x^{9} - 469905596 x^{8} + 84248201 x^{7} + 912345834 x^{6} - 152992905 x^{5} - 1188256605 x^{4} + 203114817 x^{3} + 913828487 x^{2} - 132782216 x - 329129521$	$D_{26}$ (as 26T3)	$1$	$0$
*	2.55199.26t3.b.b	$2$	$ 17^{2} \cdot 191 $	$x^{26} - x^{25} - 50 x^{24} + 48 x^{23} + 1188 x^{22} - 1066 x^{21} - 17960 x^{20} + 14231 x^{19} + 194023 x^{18} - 126696 x^{17} - 1579415 x^{16} + 794998 x^{15} + 9883811 x^{14} - 3670079 x^{13} - 47524550 x^{12} + 13070104 x^{11} + 173418938 x^{10} - 37145018 x^{9} - 469905596 x^{8} + 84248201 x^{7} + 912345834 x^{6} - 152992905 x^{5} - 1188256605 x^{4} + 203114817 x^{3} + 913828487 x^{2} - 132782216 x - 329129521$	$D_{26}$ (as 26T3)	$1$	$0$
*	2.191.13t2.a.f	$2$	$ 191 $	$x^{13} - 2 x^{12} + 4 x^{10} - 5 x^{9} + x^{8} + 5 x^{7} - 11 x^{6} + 19 x^{5} - 22 x^{4} + 16 x^{3} - 10 x^{2} + 6 x - 1$	$D_{13}$ (as 13T2)	$1$	$0$
*	2.191.13t2.a.d	$2$	$ 191 $	$x^{13} - 2 x^{12} + 4 x^{10} - 5 x^{9} + x^{8} + 5 x^{7} - 11 x^{6} + 19 x^{5} - 22 x^{4} + 16 x^{3} - 10 x^{2} + 6 x - 1$	$D_{13}$ (as 13T2)	$1$	$0$
*	2.191.13t2.a.e	$2$	$ 191 $	$x^{13} - 2 x^{12} + 4 x^{10} - 5 x^{9} + x^{8} + 5 x^{7} - 11 x^{6} + 19 x^{5} - 22 x^{4} + 16 x^{3} - 10 x^{2} + 6 x - 1$	$D_{13}$ (as 13T2)	$1$	$0$
*	2.55199.26t3.b.d	$2$	$ 17^{2} \cdot 191 $	$x^{26} - x^{25} - 50 x^{24} + 48 x^{23} + 1188 x^{22} - 1066 x^{21} - 17960 x^{20} + 14231 x^{19} + 194023 x^{18} - 126696 x^{17} - 1579415 x^{16} + 794998 x^{15} + 9883811 x^{14} - 3670079 x^{13} - 47524550 x^{12} + 13070104 x^{11} + 173418938 x^{10} - 37145018 x^{9} - 469905596 x^{8} + 84248201 x^{7} + 912345834 x^{6} - 152992905 x^{5} - 1188256605 x^{4} + 203114817 x^{3} + 913828487 x^{2} - 132782216 x - 329129521$	$D_{26}$ (as 26T3)	$1$	$0$
*	2.191.13t2.a.c	$2$	$ 191 $	$x^{13} - 2 x^{12} + 4 x^{10} - 5 x^{9} + x^{8} + 5 x^{7} - 11 x^{6} + 19 x^{5} - 22 x^{4} + 16 x^{3} - 10 x^{2} + 6 x - 1$	$D_{13}$ (as 13T2)	$1$	$0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.