Number field 28.0.537788304782666838980279974392697944146481853517.1

• Label: 28.0.537...517.1
• Degree: $28$
• Signature: $[0, 14]$
• Discriminant: $5.378\times 10^{47}$
• Root discriminant: $50.66$
• Ramified primes: $19, 197$
• Class number: $29$ (GRH)
• Class group: $[29]$ (GRH)
• Galois group: $D_{28}$ (as 28T10)

Normalized defining polynomial

\( x^{28} - 24 x^{26} + 286 x^{24} - 2327 x^{22} + 12825 x^{20} - 40336 x^{18} + 39135 x^{16} + 152588 x^{14} - 531606 x^{12} + 381077 x^{10} + 2209555 x^{8} - 10321444 x^{6} + 18143914 x^{4} - 8922944 x^{2} + 1423325 \)

Invariants

Degree:		$28$
Signature:		$[0, 14]$
Discriminant:		\(537788304782666838980279974392697944146481853517\)\(\medspace = 19^{14}\cdot 197^{13}\)
Root discriminant:		$50.66$
Ramified primes:		$19, 197$
$|\Aut(K/\Q)|$:		$2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{19} - \frac{1}{10} a^{17} - \frac{1}{5} a^{15} + \frac{3}{10} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} + \frac{2}{5} a^{7} + \frac{1}{10} a^{5} - \frac{1}{2} a^{4} - \frac{3}{10} a^{3} - \frac{1}{2} a^{2} - \frac{3}{10} a - \frac{1}{2}$, $\frac{1}{190} a^{20} + \frac{2}{95} a^{18} + \frac{1}{5} a^{16} - \frac{27}{190} a^{14} - \frac{5}{38} a^{12} - \frac{1}{2} a^{11} - \frac{3}{38} a^{10} - \frac{23}{95} a^{8} + \frac{33}{95} a^{6} - \frac{1}{2} a^{5} - \frac{13}{190} a^{4} - \frac{1}{2} a^{3} - \frac{63}{190} a^{2} - \frac{13}{38}$, $\frac{1}{190} a^{21} + \frac{2}{95} a^{19} + \frac{1}{5} a^{17} - \frac{27}{190} a^{15} - \frac{5}{38} a^{13} - \frac{1}{2} a^{12} - \frac{3}{38} a^{11} - \frac{23}{95} a^{9} + \frac{33}{95} a^{7} - \frac{1}{2} a^{6} - \frac{13}{190} a^{5} - \frac{1}{2} a^{4} - \frac{63}{190} a^{3} - \frac{13}{38} a$, $\frac{1}{190} a^{22} + \frac{11}{95} a^{18} + \frac{11}{190} a^{16} - \frac{6}{95} a^{14} - \frac{1}{2} a^{13} + \frac{17}{38} a^{12} - \frac{1}{2} a^{11} - \frac{81}{190} a^{10} - \frac{7}{38} a^{8} - \frac{1}{2} a^{7} - \frac{87}{190} a^{6} + \frac{42}{95} a^{4} - \frac{3}{190} a^{2} - \frac{5}{38}$, $\frac{1}{190} a^{23} + \frac{3}{190} a^{19} + \frac{3}{19} a^{17} + \frac{13}{95} a^{15} + \frac{14}{95} a^{13} - \frac{1}{2} a^{12} - \frac{81}{190} a^{11} - \frac{7}{38} a^{9} + \frac{27}{190} a^{7} - \frac{3}{19} a^{5} + \frac{27}{95} a^{3} - \frac{1}{2} a^{2} + \frac{16}{95} a$, $\frac{1}{334970} a^{24} - \frac{92}{167485} a^{22} - \frac{436}{167485} a^{20} + \frac{79787}{334970} a^{18} - \frac{37574}{167485} a^{16} - \frac{27872}{167485} a^{14} - \frac{1}{2} a^{13} + \frac{51242}{167485} a^{12} + \frac{76412}{167485} a^{10} + \frac{64541}{167485} a^{8} - \frac{143707}{334970} a^{6} - \frac{2536}{167485} a^{4} + \frac{39369}{334970} a^{2} - \frac{1}{2} a - \frac{6553}{66994}$, $\frac{1}{5694490} a^{25} + \frac{12157}{5694490} a^{23} - \frac{3962}{2847245} a^{21} + \frac{10804}{2847245} a^{19} - \frac{91081}{569449} a^{17} - \frac{917851}{5694490} a^{15} + \frac{42335}{569449} a^{13} + \frac{598863}{5694490} a^{11} - \frac{1}{2} a^{10} + \frac{261997}{2847245} a^{9} - \frac{1}{2} a^{8} + \frac{397986}{2847245} a^{7} - \frac{1}{2} a^{6} + \frac{98926}{569449} a^{5} - \frac{1}{2} a^{4} + \frac{1686011}{5694490} a^{3} - \frac{410413}{2847245} a - \frac{1}{2}$, $\frac{1}{253552965241621004964380111454148405250} a^{26} - \frac{340363725244844817005532862612233}{253552965241621004964380111454148405250} a^{24} - \frac{419613924332448198636226769125933117}{253552965241621004964380111454148405250} a^{22} + \frac{2633447384449397485594035049825188}{126776482620810502482190055727074202625} a^{20} + \frac{25056292054960139534428471592787307483}{126776482620810502482190055727074202625} a^{18} - \frac{4448161023512888831126016279933850238}{25355296524162100496438011145414840525} a^{16} + \frac{556354825217861600438421538064986578}{25355296524162100496438011145414840525} a^{14} - \frac{1}{2} a^{13} + \frac{55057370409198503764650882195858946484}{126776482620810502482190055727074202625} a^{12} + \frac{81017613544042945187289612317669711057}{253552965241621004964380111454148405250} a^{10} - \frac{1}{2} a^{9} - \frac{3109647191674111385461500038973138419}{13344892907453737103388426918639389750} a^{8} + \frac{100422110159481816919766471295153459429}{253552965241621004964380111454148405250} a^{6} - \frac{1}{2} a^{5} + \frac{2028070283834097855820065459792680809}{50710593048324200992876022290829681050} a^{4} - \frac{99679688775783833707458355095145573091}{253552965241621004964380111454148405250} a^{2} - \frac{1}{2} a - \frac{125224855574878995984347408367887469}{596595212333225894033835556362702130}$, $\frac{1}{253552965241621004964380111454148405250} a^{27} + \frac{15844420161147298129455677473567}{253552965241621004964380111454148405250} a^{25} - \frac{46329686433961592978349581447339721}{126776482620810502482190055727074202625} a^{23} - \frac{74173933968717652340387868906175437}{126776482620810502482190055727074202625} a^{21} + \frac{5764447376783381989478910577054961341}{253552965241621004964380111454148405250} a^{19} + \frac{8687527937591981959468663358214228579}{50710593048324200992876022290829681050} a^{17} - \frac{166151648801748138913528154618944471}{2668978581490747420677685383727877950} a^{15} + \frac{56738597691981591789856230981858659793}{253552965241621004964380111454148405250} a^{13} + \frac{871680250876026726912042986070662776}{3092109332214890304443659895782297625} a^{11} - \frac{1}{2} a^{10} + \frac{18139512460938671806489583600386059289}{253552965241621004964380111454148405250} a^{9} - \frac{1}{2} a^{8} + \frac{122393919088486925561580401918995817929}{253552965241621004964380111454148405250} a^{7} + \frac{1689009504045045811977883327148171752}{25355296524162100496438011145414840525} a^{5} - \frac{55819710891904097361064993601046909408}{126776482620810502482190055727074202625} a^{3} - \frac{2294776469627856106050399325367152049}{5071059304832420099287602229082968105} a$

Class group and class number

$C_{29}$, which has order $29$ (assuming GRH)

Unit group

Rank:		$13$
Torsion generator:		\( -1 \) (order $2$)
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:		\( 337496841957.8353 \) (assuming GRH)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 337496841957.8353 \cdot 29}{2\sqrt{537788304782666838980279974392697944146481853517}}\approx 0.997355979843452$ (assuming GRH)

Galois group

$D_{28}$ (as 28T10):

A solvable group of order 56

The 17 conjugacy class representatives for $D_{28}$

Character table for $D_{28}$

Intermediate fields

\(\Q(\sqrt{-19}) \), 4.0.71117.1, 7.1.52439613407.1, 14.0.52248348031236668805331.1

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

Sibling fields

Degree 28 sibling:

28.2.5576015581167650909427113418703236578781943428571.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$28$	$28$	${\href{/LocalNumberField/5.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$	${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$	$28$	${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$	R	${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{14}$	$28$	${\href{/LocalNumberField/37.2.0.1}{2} }^{14}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{14}$	${\href{/LocalNumberField/47.7.0.1}{7} }^{4}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{14}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
$19$	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$197$	$\Q_{197}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{197}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	197.2.1.1	$x^{2} - 197$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	197.2.1.1	$x^{2} - 197$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	197.2.1.1	$x^{2} - 197$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	197.2.1.1	$x^{2} - 197$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	197.2.1.1	$x^{2} - 197$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	197.2.1.1	$x^{2} - 197$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	197.2.1.1	$x^{2} - 197$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	197.2.1.1	$x^{2} - 197$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	197.2.1.1	$x^{2} - 197$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	197.2.1.1	$x^{2} - 197$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	197.2.1.1	$x^{2} - 197$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	197.2.1.1	$x^{2} - 197$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	197.2.1.1	$x^{2} - 197$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
	1.197.2t1.a.a	$1$	$ 197 $	\(\Q(\sqrt{197}) \)	$C_2$ (as 2T1)	$1$	$1$
	1.3743.2t1.a.a	$1$	$ 19 \cdot 197 $	\(\Q(\sqrt{-3743}) \)	$C_2$ (as 2T1)	$1$	$-1$
*	1.19.2t1.a.a	$1$	$ 19 $	\(\Q(\sqrt{-19}) \)	$C_2$ (as 2T1)	$1$	$-1$
*	2.3743.4t3.c.a	$2$	$ 19 \cdot 197 $	4.2.737371.1	$D_{4}$ (as 4T3)	$1$	$0$
*	2.3743.7t2.a.a	$2$	$ 19 \cdot 197 $	7.1.52439613407.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.3743.14t3.b.c	$2$	$ 19 \cdot 197 $	14.2.541732871692295987086853.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.3743.14t3.b.a	$2$	$ 19 \cdot 197 $	14.2.541732871692295987086853.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.3743.7t2.a.c	$2$	$ 19 \cdot 197 $	7.1.52439613407.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.3743.7t2.a.b	$2$	$ 19 \cdot 197 $	7.1.52439613407.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.3743.14t3.b.b	$2$	$ 19 \cdot 197 $	14.2.541732871692295987086853.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.3743.28t10.b.c	$2$	$ 19 \cdot 197 $	28.0.537788304782666838980279974392697944146481853517.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.3743.28t10.b.a	$2$	$ 19 \cdot 197 $	28.0.537788304782666838980279974392697944146481853517.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.3743.28t10.b.d	$2$	$ 19 \cdot 197 $	28.0.537788304782666838980279974392697944146481853517.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.3743.28t10.b.f	$2$	$ 19 \cdot 197 $	28.0.537788304782666838980279974392697944146481853517.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.3743.28t10.b.e	$2$	$ 19 \cdot 197 $	28.0.537788304782666838980279974392697944146481853517.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.3743.28t10.b.b	$2$	$ 19 \cdot 197 $	28.0.537788304782666838980279974392697944146481853517.1	$D_{28}$ (as 28T10)	$1$	$0$

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 *.