Properties

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)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(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)
 
gp: K = bnfinit(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, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4807552, 0, -11580416, 0, -4576768, 0, 503040, 0, 937984, 0, -456704, 0, -154080, 0, 19424, 0, 3424, 0, 480, 0, 656, 0, -16, 0, -30, 0, -4, 0, 1]);
 

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

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-176063041267001389140042149944365540215401480192\)\(\medspace = -\,2^{77}\cdot 71^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $48.68$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 71$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

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

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

not computed

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data 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 *.