Number field 28.2.176063041267001389140042149944365540215401480192.1

• Label: 28.2.176...192.1
• Degree: $28$
• Signature: $[2, 13]$
• Discriminant: $-1.761\times 10^{47}$
• Root discriminant: $48.68$
• Ramified primes: $2, 71$
• Class number: not computed
• Class group: not computed
• Galois group: $D_{28}$ (as 28T10)

Normalized defining polynomial

\( x^{28} - 4 x^{26} - 30 x^{24} - 16 x^{22} + 656 x^{20} + 480 x^{18} + 3424 x^{16} + 19424 x^{14} - 154080 x^{12} - 456704 x^{10} + 937984 x^{8} + 503040 x^{6} - 4576768 x^{4} - 11580416 x^{2} - 4807552 \)

Invariants

Degree:		$28$
Signature:		$[2, 13]$
Discriminant:		\(-176063041267001389140042149944365540215401480192\)\(\medspace = -\,2^{77}\cdot 71^{13}\)
Root discriminant:		$48.68$
Ramified primes:		$2, 71$
$|\Aut(K/\Q)|$:		$2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{8} a^{14}$, $\frac{1}{8} a^{15}$, $\frac{1}{16} a^{16}$, $\frac{1}{16} a^{17}$, $\frac{1}{16} a^{18}$, $\frac{1}{16} a^{19}$, $\frac{1}{32} a^{20}$, $\frac{1}{32} a^{21}$, $\frac{1}{2912} a^{22} - \frac{1}{1456} a^{20} - \frac{5}{208} a^{18} - \frac{17}{728} a^{16} + \frac{3}{91} a^{14} + \frac{45}{728} a^{12} - \frac{1}{91} a^{10} - \frac{9}{182} a^{8} - \frac{23}{182} a^{6} + \frac{3}{182} a^{4} - \frac{9}{91} a^{2} - \frac{6}{91}$, $\frac{1}{2912} a^{23} - \frac{1}{1456} a^{21} - \frac{5}{208} a^{19} - \frac{17}{728} a^{17} + \frac{3}{91} a^{15} + \frac{45}{728} a^{13} - \frac{1}{91} a^{11} - \frac{9}{182} a^{9} - \frac{23}{182} a^{7} + \frac{3}{182} a^{5} - \frac{9}{91} a^{3} - \frac{6}{91} a$, $\frac{1}{64064} a^{24} + \frac{3}{32032} a^{22} + \frac{115}{16016} a^{20} + \frac{207}{16016} a^{18} - \frac{29}{2288} a^{16} - \frac{109}{8008} a^{14} - \frac{461}{8008} a^{12} - \frac{120}{1001} a^{10} + \frac{90}{1001} a^{8} - \frac{227}{2002} a^{6} - \frac{179}{2002} a^{4} - \frac{31}{77} a^{2} + \frac{340}{1001}$, $\frac{1}{1473472} a^{25} + \frac{1}{52624} a^{23} - \frac{377}{28336} a^{21} - \frac{2547}{92092} a^{19} + \frac{1107}{92092} a^{17} + \frac{10165}{184184} a^{15} + \frac{11045}{184184} a^{13} + \frac{739}{46046} a^{11} + \frac{5167}{92092} a^{9} - \frac{5245}{23023} a^{7} - \frac{73}{23023} a^{5} - \frac{502}{23023} a^{3} - \frac{2729}{23023} a$, $\frac{1}{31072587482850164757602754982740416} a^{26} + \frac{232588595896330740523074175}{231884981215299736997035484945824} a^{24} + \frac{345190813779755627394039258665}{3884073435356270594700344372842552} a^{22} - \frac{3492231388720837219268480851305}{2219470534489297482685911070195744} a^{20} - \frac{1644052341521095943812560313371}{165279720653458323178738058418832} a^{18} - \frac{5794794727369111634131578280267}{1109735267244648741342955535097872} a^{16} - \frac{123484906580180305161841216105}{7512714575157196508124457200856} a^{14} + \frac{76685401588874098690457217908507}{3884073435356270594700344372842552} a^{12} + \frac{31141371268059151237690029962980}{485509179419533824337543046605319} a^{10} - \frac{2316552305131545879769451028427}{44137198129048529485231186055029} a^{8} + \frac{28579434918584516972733299908571}{971018358839067648675086093210638} a^{6} - \frac{917400758252157927362003628814}{44137198129048529485231186055029} a^{4} + \frac{7128613927912634488266257361216}{28559363495266695549267238035607} a^{2} + \frac{863169067047532598000892529}{22870091828137633630295494211}$, $\frac{1}{31072587482850164757602754982740416} a^{27} + \frac{150430822380416474697828891}{463769962430599473994070969891648} a^{25} + \frac{542765580305168346810716691059}{7768146870712541189400688745685104} a^{23} + \frac{26037003206933635923468266518253}{2219470534489297482685911070195744} a^{21} + \frac{2927109632421899638173343291641}{165279720653458323178738058418832} a^{19} - \frac{19134465283332997277473581682879}{1109735267244648741342955535097872} a^{17} + \frac{339292789030317649378586626399}{6356912332825320122259156093032} a^{15} - \frac{78115816485864194841197203271689}{1942036717678135297350172186421276} a^{13} + \frac{46698705547409815476055718552493}{971018358839067648675086093210638} a^{11} - \frac{210890032306243705212574210442539}{1942036717678135297350172186421276} a^{9} - \frac{117858276843470574776510190663889}{485509179419533824337543046605319} a^{7} - \frac{8551983847978446874932409442985}{485509179419533824337543046605319} a^{5} + \frac{7751330362548602049649750107334}{28559363495266695549267238035607} a^{3} + \frac{97149022467087302456730758438}{621650677873922950496213888099} a$

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

\(\Q(\sqrt{2}) \), 4.2.145408.1, 7.1.357911.1, 14.2.268645766625492992.1

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	$28$	$28$	${\href{/LocalNumberField/7.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/11.2.0.1}{2} }^{14}$	${\href{/LocalNumberField/13.2.0.1}{2} }^{14}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$	$28$	${\href{/LocalNumberField/23.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$	$28$	${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$	$28$	${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$	$28$	${\href{/LocalNumberField/47.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{14}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
2	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	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
*	1.8.2t1.a.a	$1$	$ 2^{3}$	\(\Q(\sqrt{2}) \)	$C_2$ (as 2T1)	$1$	$1$
	1.71.2t1.a.a	$1$	$ 71 $	\(\Q(\sqrt{-71}) \)	$C_2$ (as 2T1)	$1$	$-1$
	1.568.2t1.b.a	$1$	$ 2^{3} \cdot 71 $	\(\Q(\sqrt{-142}) \)	$C_2$ (as 2T1)	$1$	$-1$
*	2.18176.4t3.e.a	$2$	$ 2^{8} \cdot 71 $	4.2.145408.1	$D_{4}$ (as 4T3)	$1$	$0$
*	2.71.7t2.a.a	$2$	$ 71 $	7.1.357911.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.4544.14t3.b.c	$2$	$ 2^{6} \cdot 71 $	14.2.268645766625492992.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.4544.14t3.b.a	$2$	$ 2^{6} \cdot 71 $	14.2.268645766625492992.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.71.7t2.a.c	$2$	$ 71 $	7.1.357911.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.71.7t2.a.b	$2$	$ 71 $	7.1.357911.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.4544.14t3.b.b	$2$	$ 2^{6} \cdot 71 $	14.2.268645766625492992.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.18176.28t10.a.c	$2$	$ 2^{8} \cdot 71 $	28.2.176063041267001389140042149944365540215401480192.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.18176.28t10.a.a	$2$	$ 2^{8} \cdot 71 $	28.2.176063041267001389140042149944365540215401480192.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.18176.28t10.a.d	$2$	$ 2^{8} \cdot 71 $	28.2.176063041267001389140042149944365540215401480192.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.18176.28t10.a.f	$2$	$ 2^{8} \cdot 71 $	28.2.176063041267001389140042149944365540215401480192.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.18176.28t10.a.e	$2$	$ 2^{8} \cdot 71 $	28.2.176063041267001389140042149944365540215401480192.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.18176.28t10.a.b	$2$	$ 2^{8} \cdot 71 $	28.2.176063041267001389140042149944365540215401480192.1	$D_{28}$ (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 *.