Number field 26.2.33169014017307639762747533311185137385697693.1

• Label: 26.2.331...693.1
• Degree: $26$
• Signature: $[2, 12]$
• Discriminant: $3.317\times 10^{43}$
• Root discriminant: $47.19$
• Ramified primes: $13, 263$
• Class number: not computed
• Class group: not computed
• Galois group: $D_{26}$ (as 26T3)

Normalized defining polynomial

\( x^{26} - x^{25} - 45 x^{24} + 44 x^{23} + 912 x^{22} - 836 x^{21} - 11055 x^{20} + 9149 x^{19} + 89968 x^{18} - 63863 x^{17} - 520434 x^{16} + 297998 x^{15} + 2198217 x^{14} - 952653 x^{13} - 6808591 x^{12} + 2140242 x^{11} + 15236220 x^{10} - 3364016 x^{9} - 23680109 x^{8} + 2956997 x^{7} + 23956149 x^{6} - 618594 x^{5} - 14015975 x^{4} - 2419867 x^{3} + 3438259 x^{2} + 3561247 x - 1990843 \)

Invariants

Degree:		$26$
Signature:		$[2, 12]$
Discriminant:		\(33169014017307639762747533311185137385697693\)\(\medspace = 13^{13}\cdot 263^{12}\)
Root discriminant:		$47.19$
Ramified primes:		$13, 263$
$|\Aut(K/\Q)|$:		$2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{5} a^{19} + \frac{2}{5} a^{17} + \frac{1}{5} a^{16} - \frac{2}{5} a^{15} - \frac{2}{5} a^{14} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{35} a^{20} - \frac{2}{35} a^{19} - \frac{13}{35} a^{18} + \frac{12}{35} a^{17} + \frac{16}{35} a^{16} + \frac{2}{35} a^{15} - \frac{1}{35} a^{14} + \frac{1}{35} a^{13} + \frac{2}{35} a^{12} + \frac{3}{35} a^{11} - \frac{1}{35} a^{10} - \frac{17}{35} a^{9} - \frac{2}{35} a^{8} + \frac{2}{35} a^{6} - \frac{9}{35} a^{5} - \frac{17}{35} a^{4} + \frac{16}{35} a^{3} - \frac{3}{35} a - \frac{13}{35}$, $\frac{1}{35} a^{21} - \frac{3}{35} a^{19} - \frac{2}{5} a^{18} - \frac{2}{35} a^{17} + \frac{13}{35} a^{16} + \frac{2}{7} a^{15} + \frac{6}{35} a^{14} + \frac{4}{35} a^{13} - \frac{2}{5} a^{12} - \frac{9}{35} a^{11} - \frac{1}{7} a^{10} + \frac{13}{35} a^{9} - \frac{4}{35} a^{8} + \frac{9}{35} a^{7} + \frac{9}{35} a^{6} - \frac{2}{5} a^{5} + \frac{3}{35} a^{4} + \frac{11}{35} a^{3} + \frac{4}{35} a^{2} - \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{175} a^{22} + \frac{1}{175} a^{21} + \frac{1}{175} a^{20} + \frac{2}{35} a^{19} - \frac{33}{175} a^{18} - \frac{46}{175} a^{17} - \frac{53}{175} a^{16} - \frac{81}{175} a^{15} - \frac{64}{175} a^{14} - \frac{6}{175} a^{13} + \frac{11}{35} a^{12} - \frac{37}{175} a^{11} + \frac{74}{175} a^{10} - \frac{59}{175} a^{9} - \frac{38}{175} a^{8} - \frac{52}{175} a^{7} - \frac{32}{175} a^{6} + \frac{58}{175} a^{5} + \frac{86}{175} a^{4} - \frac{61}{175} a^{3} - \frac{1}{175} a^{2} + \frac{83}{175} a + \frac{83}{175}$, $\frac{1}{3325} a^{23} + \frac{4}{3325} a^{22} + \frac{24}{3325} a^{21} + \frac{8}{3325} a^{20} + \frac{17}{3325} a^{19} - \frac{2}{665} a^{18} + \frac{1599}{3325} a^{17} + \frac{177}{665} a^{16} - \frac{607}{3325} a^{15} + \frac{662}{3325} a^{14} + \frac{116}{475} a^{13} - \frac{617}{3325} a^{12} + \frac{398}{3325} a^{11} + \frac{1188}{3325} a^{10} - \frac{13}{133} a^{9} - \frac{1461}{3325} a^{8} + \frac{27}{3325} a^{7} + \frac{377}{3325} a^{6} + \frac{96}{665} a^{5} + \frac{1497}{3325} a^{4} + \frac{26}{3325} a^{3} + \frac{214}{665} a^{2} - \frac{383}{3325} a + \frac{144}{3325}$, $\frac{1}{9679075} a^{24} - \frac{118}{1935815} a^{23} + \frac{15394}{9679075} a^{22} + \frac{12333}{9679075} a^{21} + \frac{91291}{9679075} a^{20} - \frac{27287}{509425} a^{19} - \frac{193234}{9679075} a^{18} + \frac{1459348}{9679075} a^{17} - \frac{504818}{1935815} a^{16} + \frac{1683734}{9679075} a^{15} + \frac{140209}{387163} a^{14} + \frac{2167574}{9679075} a^{13} - \frac{1730419}{9679075} a^{12} + \frac{4035539}{9679075} a^{11} + \frac{4035187}{9679075} a^{10} - \frac{685719}{1935815} a^{9} + \frac{4822863}{9679075} a^{8} + \frac{3172292}{9679075} a^{7} - \frac{410434}{1935815} a^{6} + \frac{23415}{55309} a^{5} + \frac{134699}{9679075} a^{4} + \frac{684834}{1935815} a^{3} + \frac{932411}{9679075} a^{2} - \frac{3859216}{9679075} a - \frac{4049468}{9679075}$, $\frac{1}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{25} - \frac{307734204975035330222659251167593657042233738616740530752991964751}{6161150727067134928401881415157759652111494801704180449552991084828521375} a^{24} + \frac{9992756646826747310647570835708028598964213376297531163798778703383546}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{23} - \frac{236247647911777292033197862923787215993219823261548633568203304794184149}{117061863814275563639635746887997433390118401232379428541506830611741906125} a^{22} + \frac{7415778875938656337710747968462540743137758142952853824942188719389599733}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{21} + \frac{481479527901747967957352773485870934754721475542193026412146418223787447}{43128055089469944498813169906104317564780463611929263146870937593799649625} a^{20} + \frac{4664883492098946759230052906581473259600991032055809470320768997725155549}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{19} + \frac{294806524558570959280823172520539978362039537234781524257657549094612887171}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{18} - \frac{311543140950957883559334686896904398903542182668795100841499722198631196089}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{17} - \frac{40039656885315819068720500442349673168279432900787997850039480439487218682}{163886609339985789095490045643196406746165761725331199958109562856438668575} a^{16} - \frac{375728289435102623863976393948532245623493413958618684834459284471735154234}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{15} - \frac{387749968253620071132206545326449530740036563514075519374448178483228924259}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{14} + \frac{74611753243635617908550065767066417080314957583498052344109698816194884288}{163886609339985789095490045643196406746165761725331199958109562856438668575} a^{13} + \frac{49166466510091388438587976812216610563034622842668506436940362869815191437}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{12} + \frac{18381110790917858129735602914920483850293290751395460743808089337027948228}{163886609339985789095490045643196406746165761725331199958109562856438668575} a^{11} + \frac{163916050442621278054932040030633944414701880009158527166420298386960104152}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{10} + \frac{34313944492004964797960322860916882241181520710712724078675569827221850516}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{9} - \frac{122436882823262138942735361014565183439791872641787733169034737852235056138}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{8} + \frac{8929864446132202934490158847248765712684418849741615490838281344350403778}{43128055089469944498813169906104317564780463611929263146870937593799649625} a^{7} - \frac{30121004496968491349141479685566689637353493517822215147011733017702162192}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{6} - \frac{249950972457545530320112634265695372808380141477229648765385264726767123717}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{5} - \frac{70644582634321975545688658717490221101845290913131976905026974419802653451}{163886609339985789095490045643196406746165761725331199958109562856438668575} a^{4} + \frac{8570505700972802760035818787430700722289814272625465799130114961597107526}{23412372762855112727927149377599486678023680246475885708301366122348381225} a^{3} - \frac{325169439665242239476464712577142560880070968190654032913352044861288236792}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{2} - \frac{30752974495533762605004162950027094189590314507847019955991779662422352627}{819433046699928945477450228215982033730828808626655999790547814282193342875} a - \frac{9172856040377971125194314964116507162293098414842208354518491253933856812}{117061863814275563639635746887997433390118401232379428541506830611741906125}$

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

\(\Q(\sqrt{13}) \), 13.1.330928743953809.1

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	$26$	${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$	${\href{/LocalNumberField/5.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/7.2.0.1}{2} }^{13}$	$26$	R	${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/23.13.0.1}{13} }^{2}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$	$26$	$26$	${\href{/LocalNumberField/41.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{13}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{13}$

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
263	Data not computed

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
*	1.13.2t1.a.a	$1$	$ 13 $	\(\Q(\sqrt{13}) \)	$C_2$ (as 2T1)	$1$	$1$
	1.263.2t1.a.a	$1$	$ 263 $	\(\Q(\sqrt{-263}) \)	$C_2$ (as 2T1)	$1$	$-1$
	1.3419.2t1.a.a	$1$	$ 13 \cdot 263 $	\(\Q(\sqrt{-3419}) \)	$C_2$ (as 2T1)	$1$	$-1$
*	2.44447.26t3.a.e	$2$	$ 13^{2} \cdot 263 $	26.2.33169014017307639762747533311185137385697693.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.263.13t2.a.e	$2$	$ 263 $	13.1.330928743953809.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.263.13t2.a.c	$2$	$ 263 $	13.1.330928743953809.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.44447.26t3.a.c	$2$	$ 13^{2} \cdot 263 $	26.2.33169014017307639762747533311185137385697693.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.44447.26t3.a.a	$2$	$ 13^{2} \cdot 263 $	26.2.33169014017307639762747533311185137385697693.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.263.13t2.a.d	$2$	$ 263 $	13.1.330928743953809.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.263.13t2.a.b	$2$	$ 263 $	13.1.330928743953809.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.263.13t2.a.a	$2$	$ 263 $	13.1.330928743953809.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.44447.26t3.a.b	$2$	$ 13^{2} \cdot 263 $	26.2.33169014017307639762747533311185137385697693.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.44447.26t3.a.f	$2$	$ 13^{2} \cdot 263 $	26.2.33169014017307639762747533311185137385697693.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.263.13t2.a.f	$2$	$ 263 $	13.1.330928743953809.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.44447.26t3.a.d	$2$	$ 13^{2} \cdot 263 $	26.2.33169014017307639762747533311185137385697693.1	$D_{26}$ (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 *.