Number field 28.2.2443365527501925442977500495872000000000000000000000.1

• Label: 28.2.244...000.1
• Degree: $28$
• Signature: $[2, 13]$
• Discriminant: $-2.443\times 10^{51}$
• Root discriminant: $68.44$
• Ramified primes: $2, 5, 71$
• Class number: not computed
• Class group: not computed
• Galois group: $D_{28}$ (as 28T10)

Normalized defining polynomial

\( x^{28} + 10 x^{26} - 280 x^{24} - 2600 x^{22} + 24800 x^{20} + 288000 x^{18} - 1128000 x^{16} - 72400000 x^{14} - 791360000 x^{12} - 3704000000 x^{10} - 17401600000 x^{8} - 19584000000 x^{6} - 96256000000 x^{4} - 10240000000 x^{2} - 90880000000 \)

Invariants

Degree:		$28$
Signature:		$[2, 13]$
Discriminant:		\(-2443365527501925442977500495872000000000000000000000\)\(\medspace = -\,2^{42}\cdot 5^{21}\cdot 71^{13}\)
Root discriminant:		$68.44$
Ramified primes:		$2, 5, 71$
$|\Aut(K/\Q)|$:		$2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{20} a^{4}$, $\frac{1}{20} a^{5}$, $\frac{1}{40} a^{6}$, $\frac{1}{40} a^{7}$, $\frac{1}{400} a^{8}$, $\frac{1}{400} a^{9}$, $\frac{1}{800} a^{10}$, $\frac{1}{800} a^{11}$, $\frac{1}{8000} a^{12}$, $\frac{1}{8000} a^{13}$, $\frac{1}{16000} a^{14}$, $\frac{1}{16000} a^{15}$, $\frac{1}{1120000} a^{16} + \frac{3}{112000} a^{14} + \frac{1}{56000} a^{12} + \frac{1}{2800} a^{10} + \frac{3}{2800} a^{8} - \frac{1}{140} a^{6} - \frac{3}{140} a^{4} + \frac{1}{14} a^{2} + \frac{3}{7}$, $\frac{1}{1120000} a^{17} + \frac{3}{112000} a^{15} + \frac{1}{56000} a^{13} + \frac{1}{2800} a^{11} + \frac{3}{2800} a^{9} - \frac{1}{140} a^{7} - \frac{3}{140} a^{5} + \frac{1}{14} a^{3} + \frac{3}{7} a$, $\frac{1}{2240000} a^{18} - \frac{1}{56000} a^{14} + \frac{1}{28000} a^{12} + \frac{1}{5600} a^{10} + \frac{1}{2800} a^{8} - \frac{1}{280} a^{6} + \frac{1}{140} a^{4} + \frac{1}{7} a^{2} - \frac{3}{7}$, $\frac{1}{2240000} a^{19} - \frac{1}{56000} a^{15} + \frac{1}{28000} a^{13} + \frac{1}{5600} a^{11} + \frac{1}{2800} a^{9} - \frac{1}{280} a^{7} + \frac{1}{140} a^{5} + \frac{1}{7} a^{3} - \frac{3}{7} a$, $\frac{1}{22400000} a^{20} - \frac{1}{56000} a^{14} + \frac{3}{56000} a^{12} - \frac{1}{1400} a^{8} - \frac{1}{280} a^{6} + \frac{3}{140} a^{4} - \frac{1}{7}$, $\frac{1}{22400000} a^{21} - \frac{1}{56000} a^{15} + \frac{3}{56000} a^{13} - \frac{1}{1400} a^{9} - \frac{1}{280} a^{7} + \frac{3}{140} a^{5} - \frac{1}{7} a$, $\frac{1}{44800000} a^{22} - \frac{1}{56000} a^{14} + \frac{3}{56000} a^{12} - \frac{3}{5600} a^{10} - \frac{3}{2800} a^{8} - \frac{3}{280} a^{6} - \frac{1}{70} a^{4} + \frac{1}{7} a^{2} + \frac{2}{7}$, $\frac{1}{44800000} a^{23} - \frac{1}{56000} a^{15} + \frac{3}{56000} a^{13} - \frac{3}{5600} a^{11} - \frac{3}{2800} a^{9} - \frac{3}{280} a^{7} - \frac{1}{70} a^{5} + \frac{1}{7} a^{3} + \frac{2}{7} a$, $\frac{1}{40768000000} a^{24} - \frac{43}{4076800000} a^{22} - \frac{23}{2038400000} a^{20} + \frac{31}{203840000} a^{18} + \frac{1}{1274000} a^{14} + \frac{101}{5096000} a^{12} - \frac{101}{254800} a^{10} + \frac{33}{63700} a^{8} + \frac{19}{6370} a^{6} - \frac{17}{3185} a^{4} + \frac{271}{1274} a^{2} - \frac{178}{637}$, $\frac{1}{40768000000} a^{25} - \frac{43}{4076800000} a^{23} - \frac{23}{2038400000} a^{21} + \frac{31}{203840000} a^{19} + \frac{1}{1274000} a^{15} + \frac{101}{5096000} a^{13} - \frac{101}{254800} a^{11} + \frac{33}{63700} a^{9} + \frac{19}{6370} a^{7} - \frac{17}{3185} a^{5} + \frac{271}{1274} a^{3} - \frac{178}{637} a$, $\frac{1}{12013927979311236518304128000000} a^{26} - \frac{769079677129429317}{107267214100993183199144000000} a^{24} - \frac{217511656150610514179}{42906885640397273279657600000} a^{22} - \frac{6058747029483599664607}{300348199482780912957603200000} a^{20} + \frac{19826373471189467779}{150174099741390456478801600} a^{18} + \frac{381730111821791694919}{938588123383690352992510000} a^{16} + \frac{5605738803562426732347}{750870498706952282394008000} a^{14} - \frac{778541379741508756591}{15017409974139045647880160} a^{12} - \frac{6796658613818907916109}{18771762467673807059850200} a^{10} + \frac{7624096038230621431899}{37543524935347614119700400} a^{8} + \frac{2209279803260204442773}{750870498706952282394008} a^{6} + \frac{1372647680754925492917}{268168035252482957997860} a^{4} - \frac{33268151691021561836269}{187717624676738070598502} a^{2} - \frac{9170651213542823661245}{93858812338369035299251}$, $\frac{1}{12013927979311236518304128000000} a^{27} - \frac{769079677129429317}{107267214100993183199144000000} a^{25} - \frac{217511656150610514179}{42906885640397273279657600000} a^{23} - \frac{6058747029483599664607}{300348199482780912957603200000} a^{21} + \frac{19826373471189467779}{150174099741390456478801600} a^{19} + \frac{381730111821791694919}{938588123383690352992510000} a^{17} + \frac{5605738803562426732347}{750870498706952282394008000} a^{15} - \frac{778541379741508756591}{15017409974139045647880160} a^{13} - \frac{6796658613818907916109}{18771762467673807059850200} a^{11} + \frac{7624096038230621431899}{37543524935347614119700400} a^{9} + \frac{2209279803260204442773}{750870498706952282394008} a^{7} + \frac{1372647680754925492917}{268168035252482957997860} a^{5} - \frac{33268151691021561836269}{187717624676738070598502} a^{3} - \frac{9170651213542823661245}{93858812338369035299251} a$

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

\(\Q(\sqrt{5}) \), 4.2.568000.1, 7.1.357911.1, 14.2.10007834681328125.1

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	$28$	R	${\href{/LocalNumberField/7.2.0.1}{2} }^{14}$	${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/13.2.0.1}{2} }^{14}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$	${\href{/LocalNumberField/19.7.0.1}{7} }^{4}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{14}$	${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$	$28$	${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$	$28$	${\href{/LocalNumberField/47.2.0.1}{2} }^{14}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{14}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5	Data not computed
$71$	$\Q_{71}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{71}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	71.2.1.2	$x^{2} + 142$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.1.2	$x^{2} + 142$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.1.2	$x^{2} + 142$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.1.2	$x^{2} + 142$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.1.2	$x^{2} + 142$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.1.2	$x^{2} + 142$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.1.2	$x^{2} + 142$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.1.2	$x^{2} + 142$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.1.2	$x^{2} + 142$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.1.2	$x^{2} + 142$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.1.2	$x^{2} + 142$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.1.2	$x^{2} + 142$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.1.2	$x^{2} + 142$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
*	1.5.2t1.a.a	$1$	$ 5 $	\(\Q(\sqrt{5}) \)	$C_2$ (as 2T1)	$1$	$1$
	1.71.2t1.a.a	$1$	$ 71 $	\(\Q(\sqrt{-71}) \)	$C_2$ (as 2T1)	$1$	$-1$
	1.355.2t1.a.a	$1$	$ 5 \cdot 71 $	\(\Q(\sqrt{-355}) \)	$C_2$ (as 2T1)	$1$	$-1$
*	2.113600.4t3.c.a	$2$	$ 2^{6} \cdot 5^{2} \cdot 71 $	4.2.568000.1	$D_{4}$ (as 4T3)	$1$	$0$
*	2.1775.14t3.b.b	$2$	$ 5^{2} \cdot 71 $	14.2.10007834681328125.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.1775.14t3.b.a	$2$	$ 5^{2} \cdot 71 $	14.2.10007834681328125.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.71.7t2.a.a	$2$	$ 71 $	7.1.357911.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.71.7t2.a.c	$2$	$ 71 $	7.1.357911.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.1775.14t3.b.c	$2$	$ 5^{2} \cdot 71 $	14.2.10007834681328125.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.71.7t2.a.b	$2$	$ 71 $	7.1.357911.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.113600.28t10.a.b	$2$	$ 2^{6} \cdot 5^{2} \cdot 71 $	28.2.2443365527501925442977500495872000000000000000000000.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.113600.28t10.a.e	$2$	$ 2^{6} \cdot 5^{2} \cdot 71 $	28.2.2443365527501925442977500495872000000000000000000000.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.113600.28t10.a.f	$2$	$ 2^{6} \cdot 5^{2} \cdot 71 $	28.2.2443365527501925442977500495872000000000000000000000.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.113600.28t10.a.d	$2$	$ 2^{6} \cdot 5^{2} \cdot 71 $	28.2.2443365527501925442977500495872000000000000000000000.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.113600.28t10.a.a	$2$	$ 2^{6} \cdot 5^{2} \cdot 71 $	28.2.2443365527501925442977500495872000000000000000000000.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.113600.28t10.a.c	$2$	$ 2^{6} \cdot 5^{2} \cdot 71 $	28.2.2443365527501925442977500495872000000000000000000000.1	$D_{28}$ (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 *.