Number field 26.0.4634664002656599036916177705626061158525123.1

• Label: 26.0.463...123.1
• Degree: $26$
• Signature: $[0, 13]$
• Discriminant: $-4.635\times 10^{42}$
• Root discriminant: $43.75$
• Ramified primes: $3, 1093$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $D_{26}$ (as 26T3)

Normalized defining polynomial

\( x^{26} - 9 x^{24} - 20 x^{23} - 12 x^{22} + 186 x^{21} + 794 x^{20} - 27 x^{19} - 4404 x^{18} - 5528 x^{17} + 1038 x^{16} + 13671 x^{15} + 59660 x^{14} + 64299 x^{13} - 161565 x^{12} - 318218 x^{11} + 37881 x^{10} + 189642 x^{9} - 55703 x^{8} + 873384 x^{7} + 1978515 x^{6} - 188640 x^{5} - 4211379 x^{4} - 4130865 x^{3} + 1301535 x^{2} + 4678236 x + 2259171 \)

Invariants

Degree:		$26$
Signature:		$[0, 13]$
Discriminant:		\(-4634664002656599036916177705626061158525123\)\(\medspace = -\,3^{13}\cdot 1093^{12}\)
Root discriminant:		$43.75$
Ramified primes:		$3, 1093$
$|\Aut(K/\Q)|$:		$2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{5}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{6}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{7}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{8}$, $\frac{1}{27} a^{15} + \frac{1}{27} a^{13} + \frac{1}{27} a^{11} - \frac{1}{27} a^{9} - \frac{1}{27} a^{7} + \frac{8}{27} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{27} a^{16} + \frac{1}{27} a^{14} + \frac{1}{27} a^{12} - \frac{1}{27} a^{10} - \frac{1}{27} a^{8} - \frac{1}{27} a^{6}$, $\frac{1}{81} a^{17} + \frac{1}{27} a^{13} + \frac{1}{81} a^{11} + \frac{1}{27} a^{9} - \frac{4}{27} a^{7} + \frac{25}{81} a^{5} + \frac{8}{27} a^{3} + \frac{4}{9} a$, $\frac{1}{891} a^{18} + \frac{2}{891} a^{17} - \frac{4}{297} a^{16} - \frac{5}{297} a^{15} - \frac{4}{99} a^{14} - \frac{1}{33} a^{13} + \frac{34}{891} a^{12} - \frac{13}{891} a^{11} - \frac{10}{297} a^{10} + \frac{7}{297} a^{9} - \frac{2}{33} a^{8} - \frac{8}{99} a^{7} + \frac{127}{891} a^{6} - \frac{151}{891} a^{5} - \frac{148}{297} a^{4} - \frac{10}{27} a^{3} + \frac{28}{99} a^{2} - \frac{34}{99} a + \frac{2}{11}$, $\frac{1}{891} a^{19} - \frac{5}{891} a^{17} + \frac{1}{99} a^{16} - \frac{2}{297} a^{15} + \frac{5}{99} a^{14} + \frac{2}{81} a^{13} + \frac{2}{99} a^{12} + \frac{7}{891} a^{11} - \frac{2}{99} a^{10} + \frac{4}{99} a^{9} + \frac{4}{99} a^{8} - \frac{59}{891} a^{7} + \frac{10}{99} a^{6} + \frac{430}{891} a^{5} + \frac{40}{99} a^{4} - \frac{136}{297} a^{3} - \frac{8}{33} a^{2} - \frac{2}{99} a - \frac{4}{11}$, $\frac{1}{2673} a^{20} - \frac{1}{2673} a^{19} - \frac{1}{297} a^{17} + \frac{8}{891} a^{16} + \frac{1}{297} a^{15} + \frac{94}{2673} a^{14} - \frac{7}{2673} a^{13} - \frac{13}{891} a^{12} + \frac{4}{99} a^{11} - \frac{32}{891} a^{10} + \frac{8}{297} a^{9} - \frac{68}{2673} a^{8} - \frac{46}{2673} a^{7} + \frac{94}{891} a^{6} + \frac{8}{27} a^{5} + \frac{53}{297} a^{4} - \frac{7}{33} a^{3} - \frac{34}{99} a^{2} - \frac{13}{99} a + \frac{1}{11}$, $\frac{1}{8019} a^{21} + \frac{1}{8019} a^{20} - \frac{2}{8019} a^{19} + \frac{8}{2673} a^{17} + \frac{16}{2673} a^{16} + \frac{76}{8019} a^{15} - \frac{31}{729} a^{14} + \frac{100}{8019} a^{13} + \frac{145}{2673} a^{12} - \frac{65}{2673} a^{11} - \frac{64}{2673} a^{10} + \frac{166}{8019} a^{9} + \frac{421}{8019} a^{8} - \frac{854}{8019} a^{7} - \frac{388}{2673} a^{6} - \frac{98}{891} a^{5} + \frac{61}{891} a^{4} + \frac{5}{33} a^{3} + \frac{5}{11} a^{2} + \frac{112}{297} a + \frac{8}{33}$, $\frac{1}{8019} a^{22} - \frac{1}{8019} a^{19} - \frac{1}{2673} a^{18} + \frac{14}{2673} a^{17} + \frac{127}{8019} a^{16} + \frac{5}{2673} a^{15} - \frac{128}{2673} a^{14} - \frac{442}{8019} a^{13} - \frac{20}{891} a^{12} - \frac{38}{2673} a^{11} + \frac{26}{729} a^{10} + \frac{67}{2673} a^{9} - \frac{205}{2673} a^{8} - \frac{53}{729} a^{7} - \frac{371}{2673} a^{6} + \frac{62}{891} a^{5} + \frac{179}{891} a^{4} + \frac{49}{297} a^{3} + \frac{118}{297} a^{2} + \frac{62}{297} a + \frac{10}{33}$, $\frac{1}{24057} a^{23} - \frac{1}{24057} a^{22} - \frac{1}{24057} a^{20} + \frac{7}{24057} a^{19} + \frac{1}{2673} a^{18} - \frac{131}{24057} a^{17} + \frac{104}{24057} a^{16} + \frac{29}{8019} a^{15} - \frac{1327}{24057} a^{14} + \frac{1135}{24057} a^{13} - \frac{134}{8019} a^{12} - \frac{950}{24057} a^{11} + \frac{239}{24057} a^{10} + \frac{376}{8019} a^{9} - \frac{2749}{24057} a^{8} + \frac{46}{24057} a^{7} - \frac{4}{8019} a^{6} + \frac{259}{891} a^{5} - \frac{173}{2673} a^{4} + \frac{128}{297} a^{3} + \frac{229}{891} a^{2} + \frac{298}{891} a - \frac{13}{99}$, $\frac{1}{697653} a^{24} + \frac{1}{63423} a^{23} - \frac{13}{232551} a^{22} - \frac{2}{63423} a^{21} - \frac{107}{697653} a^{20} + \frac{1}{2871} a^{19} - \frac{212}{697653} a^{18} - \frac{1729}{697653} a^{17} - \frac{8}{232551} a^{16} + \frac{343}{24057} a^{15} + \frac{25339}{697653} a^{14} + \frac{8224}{232551} a^{13} + \frac{12442}{697653} a^{12} - \frac{15166}{697653} a^{11} - \frac{2855}{77517} a^{10} + \frac{11756}{697653} a^{9} + \frac{110740}{697653} a^{8} + \frac{1604}{232551} a^{7} + \frac{197}{2871} a^{6} - \frac{26393}{77517} a^{5} + \frac{1904}{8613} a^{4} + \frac{1399}{25839} a^{3} + \frac{2011}{25839} a^{2} - \frac{140}{957} a - \frac{84}{319}$, $\frac{1}{81532389886724403571936116880489530454479711} a^{25} - \frac{56139696771393469470078943515625058918}{81532389886724403571936116880489530454479711} a^{24} - \frac{1022436181847995615356088599410464433897}{81532389886724403571936116880489530454479711} a^{23} + \frac{1499627502792998190703703851200014622542}{81532389886724403571936116880489530454479711} a^{22} - \frac{4783584305954396314330410923462447008210}{81532389886724403571936116880489530454479711} a^{21} - \frac{1240232907110849259068322884638153436821}{81532389886724403571936116880489530454479711} a^{20} - \frac{8130054289359684157309011202213884334601}{81532389886724403571936116880489530454479711} a^{19} - \frac{31617025266451232047074267891935731756733}{81532389886724403571936116880489530454479711} a^{18} - \frac{231386328170740317532053382872664341451868}{81532389886724403571936116880489530454479711} a^{17} - \frac{339513753044780508099511159172908401721141}{81532389886724403571936116880489530454479711} a^{16} + \frac{964000215617017679958006920992752586473728}{81532389886724403571936116880489530454479711} a^{15} - \frac{3299818074689503809870289742284669170234304}{81532389886724403571936116880489530454479711} a^{14} - \frac{3270330785687954206341135904963896998256830}{81532389886724403571936116880489530454479711} a^{13} - \frac{195815788799725034392849305707690004073180}{7412035444247673051994192443680866404952701} a^{12} + \frac{2963533154359106565630282972025862912052952}{81532389886724403571936116880489530454479711} a^{11} - \frac{2089731924310030740353049719468331000094432}{81532389886724403571936116880489530454479711} a^{10} + \frac{22990411929013046719003884575098325818028}{1896102090388939617952002718150919312894877} a^{9} + \frac{10270045632509183114589945045557661166138604}{81532389886724403571936116880489530454479711} a^{8} + \frac{2287865854310620760587301594238991835476171}{27177463295574801190645372293496510151493237} a^{7} - \frac{640651972033487850962725561392289068767420}{9059154431858267063548457431165503383831079} a^{6} + \frac{1051186582342645786362314111254440132565424}{9059154431858267063548457431165503383831079} a^{5} - \frac{326945052818760509208787248355166575642307}{3019718143952755687849485810388501127943693} a^{4} + \frac{227301349998375404394209981847805877169974}{1006572714650918562616495270129500375981231} a^{3} - \frac{56875137735393937697153054638850113947346}{3019718143952755687849485810388501127943693} a^{2} + \frac{162213268691348358943579540551237746182118}{335524238216972854205498423376500125327077} a - \frac{8555948587850113940220145211874441253741}{37280470912996983800610935930722236147453}$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$12$
Torsion generator:		\( \frac{36208078418198207877821}{4458879115785042788098908891} a^{25} - \frac{359615535410293894115}{45039182987727704930292009} a^{24} - \frac{31262516208027746085383}{495431012865004754233212099} a^{23} - \frac{425990883981092931175735}{4458879115785042788098908891} a^{22} - \frac{22199755028563812957317}{1486293038595014262699636297} a^{21} + \frac{2169545031902070785718007}{1486293038595014262699636297} a^{20} + \frac{1966809765563692650586139}{405352646889549344372628081} a^{19} - \frac{2436372673414889562011809}{495431012865004754233212099} a^{18} - \frac{43113806312603747638121330}{1486293038595014262699636297} a^{17} - \frac{55496681384568909869166544}{4458879115785042788098908891} a^{16} + \frac{27681912439440856669971023}{1486293038595014262699636297} a^{15} + \frac{12712401746592951227598830}{165143670955001584744404033} a^{14} + \frac{1692552223897118071205295928}{4458879115785042788098908891} a^{13} + \frac{168498381437229074001685235}{1486293038595014262699636297} a^{12} - \frac{2028905847961002529244927770}{1486293038595014262699636297} a^{11} - \frac{4313244339113148598844038264}{4458879115785042788098908891} a^{10} + \frac{720907275691409161955384764}{495431012865004754233212099} a^{9} - \frac{19088612661357788993348372}{135117548963183114790876027} a^{8} - \frac{260934342760137406650214223}{405352646889549344372628081} a^{7} + \frac{10759599472953414157159779574}{1486293038595014262699636297} a^{6} + \frac{336473067120924159363559183}{45039182987727704930292009} a^{5} - \frac{4829827678413257911798921813}{495431012865004754233212099} a^{4} - \frac{3635339375171236108272860357}{165143670955001584744404033} a^{3} - \frac{1174528933481695845663433988}{165143670955001584744404033} a^{2} + \frac{3149028082692286688645040485}{165143670955001584744404033} a + \frac{296381255429909873085936859}{18349296772777953860489337} \) (order $6$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 16355807538960.088 \) (assuming GRH)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{13}\cdot 16355807538960.088 \cdot 1}{6\sqrt{4634664002656599036916177705626061158525123}}\approx 30.1196487161707$ (assuming GRH)

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

\(\Q(\sqrt{-3}) \), 13.1.1242935235998051916321.1

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	$26$	R	$26$	${\href{/LocalNumberField/7.13.0.1}{13} }^{2}$	${\href{/LocalNumberField/11.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$	$26$	${\href{/LocalNumberField/19.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{13}$	$26$	$26$

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
$3$	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
1093	Data not computed

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
	1.1093.2t1.a.a	$1$	$ 1093 $	\(\Q(\sqrt{1093}) \)	$C_2$ (as 2T1)	$1$	$1$
	1.3279.2t1.a.a	$1$	$ 3 \cdot 1093 $	\(\Q(\sqrt{-3279}) \)	$C_2$ (as 2T1)	$1$	$-1$
*	1.3.2t1.a.a	$1$	$ 3 $	\(\Q(\sqrt{-3}) \)	$C_2$ (as 2T1)	$1$	$-1$
*	2.3279.26t3.a.b	$2$	$ 3 \cdot 1093 $	26.0.4634664002656599036916177705626061158525123.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.3279.13t2.a.a	$2$	$ 3 \cdot 1093 $	13.1.1242935235998051916321.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.3279.13t2.a.f	$2$	$ 3 \cdot 1093 $	13.1.1242935235998051916321.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.3279.26t3.a.d	$2$	$ 3 \cdot 1093 $	26.0.4634664002656599036916177705626061158525123.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.3279.26t3.a.c	$2$	$ 3 \cdot 1093 $	26.0.4634664002656599036916177705626061158525123.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.3279.13t2.a.c	$2$	$ 3 \cdot 1093 $	13.1.1242935235998051916321.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.3279.13t2.a.e	$2$	$ 3 \cdot 1093 $	13.1.1242935235998051916321.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.3279.13t2.a.b	$2$	$ 3 \cdot 1093 $	13.1.1242935235998051916321.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.3279.26t3.a.f	$2$	$ 3 \cdot 1093 $	26.0.4634664002656599036916177705626061158525123.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.3279.26t3.a.e	$2$	$ 3 \cdot 1093 $	26.0.4634664002656599036916177705626061158525123.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.3279.13t2.a.d	$2$	$ 3 \cdot 1093 $	13.1.1242935235998051916321.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.3279.26t3.a.a	$2$	$ 3 \cdot 1093 $	26.0.4634664002656599036916177705626061158525123.1	$D_{26}$ (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 *.