Number field 26.2.713943735055873357083710772983186869616893.1

• Label: 26.2.713...893.1
• Degree: $26$
• Signature: $[2, 12]$
• Discriminant: $7.139\times 10^{41}$
• Root discriminant: $40.72$
• Ramified primes: $13, 191$
• Class number: not computed
• Class group: not computed
• Galois group: $D_{26}$ (as 26T3)

Normalized defining polynomial

\( x^{26} - x^{25} - 37 x^{24} + 36 x^{23} + 656 x^{22} - 602 x^{21} - 7532 x^{20} + 5987 x^{19} + 63242 x^{18} - 39168 x^{17} - 407978 x^{16} + 177750 x^{15} + 2044391 x^{14} - 593805 x^{13} - 7863783 x^{12} + 1620832 x^{11} + 22802242 x^{10} - 3804296 x^{9} - 48757198 x^{8} + 7281003 x^{7} + 74301119 x^{6} - 12442481 x^{5} - 74591007 x^{4} + 16397553 x^{3} + 41639217 x^{2} - 9412671 x - 11716513 \)

Invariants

Degree:		$26$
Signature:		$[2, 12]$
Discriminant:		\(713943735055873357083710772983186869616893\)\(\medspace = 13^{13}\cdot 191^{12}\)
Root discriminant:		$40.72$
Ramified primes:		$13, 191$
$|\Aut(K/\Q)|$:		$2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{7} a^{21} + \frac{2}{7} a^{20} - \frac{2}{7} a^{19} - \frac{1}{7} a^{18} + \frac{3}{7} a^{17} - \frac{1}{7} a^{15} + \frac{3}{7} a^{14} - \frac{3}{7} a^{13} - \frac{3}{7} a^{12} - \frac{2}{7} a^{11} - \frac{3}{7} a^{10} - \frac{2}{7} a^{7} - \frac{3}{7} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{4} + \frac{1}{7} a^{3} - \frac{2}{7}$, $\frac{1}{7} a^{22} + \frac{1}{7} a^{20} + \frac{3}{7} a^{19} - \frac{2}{7} a^{18} + \frac{1}{7} a^{17} - \frac{1}{7} a^{16} - \frac{2}{7} a^{15} - \frac{2}{7} a^{14} + \frac{3}{7} a^{13} - \frac{3}{7} a^{12} + \frac{1}{7} a^{11} - \frac{1}{7} a^{10} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} + \frac{2}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{23} + \frac{1}{7} a^{20} + \frac{2}{7} a^{18} + \frac{3}{7} a^{17} - \frac{2}{7} a^{16} - \frac{1}{7} a^{15} - \frac{3}{7} a^{12} + \frac{1}{7} a^{11} + \frac{3}{7} a^{10} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a^{2} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{407030407} a^{24} + \frac{26500680}{407030407} a^{23} - \frac{26746959}{407030407} a^{22} - \frac{16747158}{407030407} a^{21} - \frac{809895}{3602039} a^{20} - \frac{168841075}{407030407} a^{19} + \frac{121204474}{407030407} a^{18} + \frac{2485405}{8306743} a^{17} - \frac{72052441}{407030407} a^{16} - \frac{152047013}{407030407} a^{15} + \frac{61556484}{407030407} a^{14} + \frac{4123975}{58147201} a^{13} + \frac{16250571}{407030407} a^{12} + \frac{179770040}{407030407} a^{11} - \frac{414085}{8306743} a^{10} - \frac{23961327}{407030407} a^{9} - \frac{94350002}{407030407} a^{8} - \frac{2417505}{407030407} a^{7} + \frac{41840655}{407030407} a^{6} + \frac{21041534}{407030407} a^{5} + \frac{5485625}{21422653} a^{4} - \frac{85395049}{407030407} a^{3} - \frac{73695659}{407030407} a^{2} + \frac{26168305}{407030407} a - \frac{17597504}{407030407}$, $\frac{1}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{25} + \frac{2817902550531034922440596344724118625648516882205332745727943828816057507851}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{24} + \frac{25633179683937210952200413519014468635540031485672101444897935286563806296882838900}{639359059067385794816264986105677896627003001617342874252774471785176946217028355569} a^{23} + \frac{285188829684754932185736697198495335886856683284196967275062599671412895042222108250}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{22} - \frac{1528944139521844170390685906279603248340666650545833364067668312814649557297489612}{33650476793020304990329736110825152454052789558807519697514445883430365590369913451} a^{21} + \frac{240271694835107178741605553828903905427934370647819591304960433994277108556797299890}{639359059067385794816264986105677896627003001617342874252774471785176946217028355569} a^{20} - \frac{27160438997221409640914807351379884154858358699600731965088024239841130015226210674}{91337008438197970688037855157953985232429000231048982036110638826453849459575479367} a^{19} + \frac{196597662592663935607412397912609323751996643731476029114867328367660936756082149132}{406864855770154596701259536612704116035365546483763647251765572954203511229018044453} a^{18} - \frac{1225836950162667252198973362115337557400889629564884924586909968655157276916995268635}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{17} - \frac{111507807809919078537153231334440439687401118693625384511681444631979637306947065899}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{16} + \frac{1120238464320768430023543122132617228149679595278101156802977899088080943477670414656}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{15} + \frac{2191687245068664655474871902446777596249228543514073727261284509057779019884548540368}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{14} + \frac{188058963851948476218220194693495684343871835099381481859061501333764092329568457946}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{13} + \frac{111482647893157219231457639087075652191861541709712454976666882403285815532575729979}{639359059067385794816264986105677896627003001617342874252774471785176946217028355569} a^{12} + \frac{129633899200774173357995098702706190040437652830157608874181175241218438300069260022}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{11} - \frac{1367654813433467918224250935390690015795292785499024959384840687208284550380828178867}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{10} + \frac{2017268097943887318798001482822784911153281156023722418937116900002631183304424658775}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{9} - \frac{1427983676775956461361093700774150873232955778315643038511212195513690778554054169010}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{8} - \frac{1910242237873322387430769400140709413490819144258311134576346566017197365996989197612}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{7} - \frac{440842511816791535095401996371200254386108415409145172765345223996820882756314172973}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{6} + \frac{1704640218688968812449021278266973425767774868395587637628587264117715908959257219996}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{5} - \frac{2232896232029545305418650273464719051343814777827982383786483716058969796388364810982}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{4} - \frac{2089368399433922149756832433185955799843556156338278487526370626383839155456408886108}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{3} + \frac{510946582554466015771480386600272033650080704854328993474863782288222764158669481486}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{2} + \frac{6740537480707759232441605735749622760543786006925400955923025642966773484861626006}{39606313393554872245255353121590666162734699215233629378490454004391492243532729991} a + \frac{1211557338016559476502033276566541352600596217290438386907870571220906801993495147779}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983}$

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

\(\Q(\sqrt{13}) \), 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:

Deg 26

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$26$	${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$	$26$	${\href{/LocalNumberField/7.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/11.2.0.1}{2} }^{13}$	R	${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/23.13.0.1}{13} }^{2}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$	$26$

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
13	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	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
*	1.13.2t1.a.a	$1$	$ 13 $	\(\Q(\sqrt{13}) \)	$C_2$ (as 2T1)	$1$	$1$
	1.191.2t1.a.a	$1$	$ 191 $	\(\Q(\sqrt{-191}) \)	$C_2$ (as 2T1)	$1$	$-1$
	1.2483.2t1.a.a	$1$	$ 13 \cdot 191 $	\(\Q(\sqrt{-2483}) \)	$C_2$ (as 2T1)	$1$	$-1$
*	2.32279.26t3.a.d	$2$	$ 13^{2} \cdot 191 $	26.2.713943735055873357083710772983186869616893.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.191.13t2.a.f	$2$	$ 191 $	13.1.48551226272641.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.191.13t2.a.e	$2$	$ 191 $	13.1.48551226272641.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.32279.26t3.a.e	$2$	$ 13^{2} \cdot 191 $	26.2.713943735055873357083710772983186869616893.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.32279.26t3.a.f	$2$	$ 13^{2} \cdot 191 $	26.2.713943735055873357083710772983186869616893.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.191.13t2.a.b	$2$	$ 191 $	13.1.48551226272641.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.191.13t2.a.c	$2$	$ 191 $	13.1.48551226272641.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.191.13t2.a.d	$2$	$ 191 $	13.1.48551226272641.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.32279.26t3.a.a	$2$	$ 13^{2} \cdot 191 $	26.2.713943735055873357083710772983186869616893.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.32279.26t3.a.c	$2$	$ 13^{2} \cdot 191 $	26.2.713943735055873357083710772983186869616893.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.191.13t2.a.a	$2$	$ 191 $	13.1.48551226272641.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.32279.26t3.a.b	$2$	$ 13^{2} \cdot 191 $	26.2.713943735055873357083710772983186869616893.1	$D_{26}$ (as 26T3)	$1$	$0$

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 *.