Number field 28.2.3206343011700717904956462085952501156673661196828672.1

• Label: 28.2.320...672.1
• Degree: $28$
• Signature: $[2, 13]$
• Discriminant: $-3.206\times 10^{51}$
• Root discriminant: $69.10$
• Ramified primes: $2, 151$
• Class number: not computed
• Class group: not computed
• Galois group: $D_{28}$ (as 28T10)

Normalized defining polynomial

\( x^{28} - 24 x^{24} - 160 x^{22} - 1656 x^{20} + 35040 x^{18} - 27616 x^{16} - 906528 x^{14} + 2661648 x^{12} + 11464832 x^{10} - 75391104 x^{8} + 141558144 x^{6} - 105026816 x^{4} + 23099904 x^{2} - 1565568 \)

Invariants

Degree:		$28$
Signature:		$[2, 13]$
Discriminant:		\(-3206343011700717904956462085952501156673661196828672\)\(\medspace = -\,2^{77}\cdot 151^{13}\)
Root discriminant:		$69.10$
Ramified primes:		$2, 151$
$|\Aut(K/\Q)|$:		$2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{12} a^{8} - \frac{1}{6} a^{6} - \frac{1}{6} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{12} a^{9} - \frac{1}{3} a$, $\frac{1}{12} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{12} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{24} a^{12} - \frac{1}{6} a^{4}$, $\frac{1}{24} a^{13} - \frac{1}{6} a^{5}$, $\frac{1}{72} a^{14} - \frac{1}{72} a^{12} - \frac{1}{36} a^{8} - \frac{1}{6} a^{6} + \frac{1}{9} a^{4} + \frac{2}{9} a^{2}$, $\frac{1}{72} a^{15} - \frac{1}{72} a^{13} - \frac{1}{36} a^{9} - \frac{2}{9} a^{5} - \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{144} a^{16} + \frac{1}{72} a^{12} + \frac{1}{36} a^{10} + \frac{1}{36} a^{8} + \frac{2}{9} a^{6} - \frac{1}{6} a^{4} + \frac{4}{9} a^{2}$, $\frac{1}{144} a^{17} + \frac{1}{72} a^{13} + \frac{1}{36} a^{11} + \frac{1}{36} a^{9} + \frac{1}{18} a^{7} + \frac{1}{6} a^{5} - \frac{2}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{144} a^{18} + \frac{1}{36} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{144} a^{19} + \frac{1}{36} a^{11} - \frac{2}{9} a^{3}$, $\frac{1}{288} a^{20} + \frac{1}{72} a^{12} - \frac{1}{9} a^{4}$, $\frac{1}{864} a^{21} + \frac{1}{216} a^{15} - \frac{1}{72} a^{13} - \frac{1}{36} a^{11} + \frac{1}{54} a^{9} + \frac{1}{18} a^{7} + \frac{1}{6} a^{5} - \frac{1}{27} a^{3} - \frac{1}{3} a$, $\frac{1}{78624} a^{22} - \frac{5}{3744} a^{20} + \frac{3}{1456} a^{18} - \frac{103}{39312} a^{16} - \frac{5}{936} a^{14} - \frac{131}{6552} a^{12} + \frac{17}{756} a^{10} + \frac{67}{3276} a^{8} - \frac{17}{234} a^{6} + \frac{1021}{4914} a^{4} + \frac{44}{819} a^{2} + \frac{9}{91}$, $\frac{1}{78624} a^{23} - \frac{1}{5616} a^{21} + \frac{3}{1456} a^{19} - \frac{103}{39312} a^{17} - \frac{1}{1404} a^{15} + \frac{17}{2184} a^{13} - \frac{1}{189} a^{11} + \frac{383}{9828} a^{9} - \frac{2}{117} a^{7} + \frac{1021}{4914} a^{5} + \frac{41}{2457} a^{3} - \frac{64}{273} a$, $\frac{1}{1415232} a^{24} - \frac{1}{176904} a^{22} - \frac{5}{4536} a^{20} - \frac{1037}{353808} a^{18} - \frac{323}{353808} a^{16} - \frac{1}{273} a^{14} + \frac{1499}{176904} a^{12} + \frac{445}{88452} a^{10} + \frac{223}{14742} a^{8} + \frac{6401}{44226} a^{6} + \frac{1279}{22113} a^{4} - \frac{269}{819} a^{2} + \frac{100}{273}$, $\frac{1}{1415232} a^{25} - \frac{1}{176904} a^{23} + \frac{1}{18144} a^{21} - \frac{1037}{353808} a^{19} - \frac{323}{353808} a^{17} + \frac{19}{19656} a^{15} - \frac{479}{88452} a^{13} - \frac{503}{22113} a^{11} + \frac{248}{7371} a^{9} + \frac{1487}{44226} a^{7} + \frac{1279}{22113} a^{5} - \frac{79}{2457} a^{3} + \frac{100}{273} a$, $\frac{1}{101081515688727484577604090607935936} a^{26} - \frac{9937481208340290893208335125}{101081515688727484577604090607935936} a^{24} - \frac{82966062596966047332367048817}{25270378922181871144401022651983984} a^{22} - \frac{2703041600651213493328674421115}{3610054131740267306343003235997712} a^{20} + \frac{2892143457537796605422602121555}{3610054131740267306343003235997712} a^{18} - \frac{68418116557086139107127404848705}{25270378922181871144401022651983984} a^{16} + \frac{79510949957139134754136137127169}{12635189461090935572200511325991992} a^{14} + \frac{238883814099308563993639536724987}{12635189461090935572200511325991992} a^{12} - \frac{7943766112218449192069163952793}{6317594730545467786100255662995996} a^{10} - \frac{17586658429596916673624468797945}{3158797365272733893050127831497998} a^{8} + \frac{30467189702287412084156137616029}{242984412713287222542317525499846} a^{6} - \frac{268527667271240114703609135179071}{3158797365272733893050127831497998} a^{4} - \frac{137695289971381110203566924220}{6499583056116736405452937924893} a^{2} - \frac{676273727539078448743788132626}{1499903782180785324335293367283}$, $\frac{1}{101081515688727484577604090607935936} a^{27} - \frac{9937481208340290893208335125}{101081515688727484577604090607935936} a^{25} - \frac{82966062596966047332367048817}{25270378922181871144401022651983984} a^{23} + \frac{2950523585133376963210714204061}{7220108263480534612686006471995424} a^{21} + \frac{2892143457537796605422602121555}{3610054131740267306343003235997712} a^{19} - \frac{68418116557086139107127404848705}{25270378922181871144401022651983984} a^{17} - \frac{37481545052962120544016745520905}{12635189461090935572200511325991992} a^{15} + \frac{238883814099308563993639536724987}{12635189461090935572200511325991992} a^{13} - \frac{45858127156842583034824621981226}{1579398682636366946525063915748999} a^{11} + \frac{257307920666059304898133269024295}{6317594730545467786100255662995996} a^{9} + \frac{3468921623033276246120857004935}{242984412713287222542317525499846} a^{7} + \frac{433427302789367417085308160709373}{3158797365272733893050127831497998} a^{5} + \frac{7530848976450756720498445579987}{19498749168350209216358813774679} a^{3} - \frac{176305800145483340632023676865}{1499903782180785324335293367283} a$

Class group and class number

not computed

Unit group

Rank:		$14$
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

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

A solvable group of order 56

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

Character table for $D_{28}$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.2.309248.4, 7.1.3442951.1, 14.2.24859454395438333952.1

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

Sibling fields

Degree 28 sibling:

Deg 28

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/LocalNumberField/3.2.0.1}{2} }^{14}$	$28$	${\href{/LocalNumberField/7.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$	$28$	${\href{/LocalNumberField/13.2.0.1}{2} }^{14}$	${\href{/LocalNumberField/17.7.0.1}{7} }^{4}$	$28$	${\href{/LocalNumberField/23.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$	$28$	${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$	$28$	${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$	$28$	${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{14}$	$28$

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
2	Data not computed
$151$	$\Q_{151}$	$x + 5$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{151}$	$x + 5$	$1$	$1$	$0$	Trivial	$[\ ]$
	151.2.1.2	$x^{2} + 755$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	151.2.1.2	$x^{2} + 755$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	151.2.1.2	$x^{2} + 755$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	151.2.1.2	$x^{2} + 755$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	151.2.1.2	$x^{2} + 755$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	151.2.1.2	$x^{2} + 755$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	151.2.1.2	$x^{2} + 755$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	151.2.1.2	$x^{2} + 755$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	151.2.1.2	$x^{2} + 755$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	151.2.1.2	$x^{2} + 755$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	151.2.1.2	$x^{2} + 755$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	151.2.1.2	$x^{2} + 755$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	151.2.1.2	$x^{2} + 755$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.8.2t1.a.a	$1$	$ 2^{3}$	$x^{2} - 2$	$C_2$ (as 2T1)	$1$	$1$
	1.151.2t1.a.a	$1$	$ 151 $	$x^{2} - x + 38$	$C_2$ (as 2T1)	$1$	$-1$
	1.1208.2t1.b.a	$1$	$ 2^{3} \cdot 151 $	$x^{2} + 302$	$C_2$ (as 2T1)	$1$	$-1$
*	2.38656.4t3.a.a	$2$	$ 2^{8} \cdot 151 $	$x^{4} + 12 x^{2} - 302$	$D_{4}$ (as 4T3)	$1$	$0$
*	2.151.7t2.a.c	$2$	$ 151 $	$x^{7} - x^{6} + x^{5} + 3 x^{3} - x^{2} + 3 x + 1$	$D_{7}$ (as 7T2)	$1$	$0$
*	2.9664.14t3.a.a	$2$	$ 2^{6} \cdot 151 $	$x^{14} + 2 x^{12} + 28 x^{10} + 80 x^{8} + 272 x^{6} + 544 x^{4} + 704 x^{2} - 128$	$D_{14}$ (as 14T3)	$1$	$0$
*	2.9664.14t3.a.b	$2$	$ 2^{6} \cdot 151 $	$x^{14} + 2 x^{12} + 28 x^{10} + 80 x^{8} + 272 x^{6} + 544 x^{4} + 704 x^{2} - 128$	$D_{14}$ (as 14T3)	$1$	$0$
*	2.151.7t2.a.a	$2$	$ 151 $	$x^{7} - x^{6} + x^{5} + 3 x^{3} - x^{2} + 3 x + 1$	$D_{7}$ (as 7T2)	$1$	$0$
*	2.151.7t2.a.b	$2$	$ 151 $	$x^{7} - x^{6} + x^{5} + 3 x^{3} - x^{2} + 3 x + 1$	$D_{7}$ (as 7T2)	$1$	$0$
*	2.9664.14t3.a.c	$2$	$ 2^{6} \cdot 151 $	$x^{14} + 2 x^{12} + 28 x^{10} + 80 x^{8} + 272 x^{6} + 544 x^{4} + 704 x^{2} - 128$	$D_{14}$ (as 14T3)	$1$	$0$
*	2.38656.28t10.a.b	$2$	$ 2^{8} \cdot 151 $	$x^{28} - 24 x^{24} - 160 x^{22} - 1656 x^{20} + 35040 x^{18} - 27616 x^{16} - 906528 x^{14} + 2661648 x^{12} + 11464832 x^{10} - 75391104 x^{8} + 141558144 x^{6} - 105026816 x^{4} + 23099904 x^{2} - 1565568$	$D_{28}$ (as 28T10)	$1$	$0$
*	2.38656.28t10.a.f	$2$	$ 2^{8} \cdot 151 $	$x^{28} - 24 x^{24} - 160 x^{22} - 1656 x^{20} + 35040 x^{18} - 27616 x^{16} - 906528 x^{14} + 2661648 x^{12} + 11464832 x^{10} - 75391104 x^{8} + 141558144 x^{6} - 105026816 x^{4} + 23099904 x^{2} - 1565568$	$D_{28}$ (as 28T10)	$1$	$0$
*	2.38656.28t10.a.c	$2$	$ 2^{8} \cdot 151 $	$x^{28} - 24 x^{24} - 160 x^{22} - 1656 x^{20} + 35040 x^{18} - 27616 x^{16} - 906528 x^{14} + 2661648 x^{12} + 11464832 x^{10} - 75391104 x^{8} + 141558144 x^{6} - 105026816 x^{4} + 23099904 x^{2} - 1565568$	$D_{28}$ (as 28T10)	$1$	$0$
*	2.38656.28t10.a.d	$2$	$ 2^{8} \cdot 151 $	$x^{28} - 24 x^{24} - 160 x^{22} - 1656 x^{20} + 35040 x^{18} - 27616 x^{16} - 906528 x^{14} + 2661648 x^{12} + 11464832 x^{10} - 75391104 x^{8} + 141558144 x^{6} - 105026816 x^{4} + 23099904 x^{2} - 1565568$	$D_{28}$ (as 28T10)	$1$	$0$
*	2.38656.28t10.a.a	$2$	$ 2^{8} \cdot 151 $	$x^{28} - 24 x^{24} - 160 x^{22} - 1656 x^{20} + 35040 x^{18} - 27616 x^{16} - 906528 x^{14} + 2661648 x^{12} + 11464832 x^{10} - 75391104 x^{8} + 141558144 x^{6} - 105026816 x^{4} + 23099904 x^{2} - 1565568$	$D_{28}$ (as 28T10)	$1$	$0$
*	2.38656.28t10.a.e	$2$	$ 2^{8} \cdot 151 $	$x^{28} - 24 x^{24} - 160 x^{22} - 1656 x^{20} + 35040 x^{18} - 27616 x^{16} - 906528 x^{14} + 2661648 x^{12} + 11464832 x^{10} - 75391104 x^{8} + 141558144 x^{6} - 105026816 x^{4} + 23099904 x^{2} - 1565568$	$D_{28}$ (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 *.