Properties

Label 28.0.650...777.1
Degree $28$
Signature $[0, 14]$
Discriminant $6.508\times 10^{41}$
Root discriminant $31.14$
Ramified primes $7, 71$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{28}$ (as 28T10)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 2*x^27 + 4*x^26 - 10*x^25 + 5*x^24 + 7*x^23 - 18*x^22 - 225*x^21 - 120*x^20 + 1331*x^19 - 2152*x^18 + 4025*x^17 + 6967*x^16 - 15748*x^15 + 11604*x^14 - 4171*x^13 - 26180*x^12 + 37520*x^11 - 9040*x^10 + 16909*x^9 - 40976*x^8 - 11561*x^7 + 79270*x^6 - 49005*x^5 + 81626*x^4 - 70530*x^3 + 35205*x^2 - 15236*x + 3229)
 
gp: K = bnfinit(x^28 - 2*x^27 + 4*x^26 - 10*x^25 + 5*x^24 + 7*x^23 - 18*x^22 - 225*x^21 - 120*x^20 + 1331*x^19 - 2152*x^18 + 4025*x^17 + 6967*x^16 - 15748*x^15 + 11604*x^14 - 4171*x^13 - 26180*x^12 + 37520*x^11 - 9040*x^10 + 16909*x^9 - 40976*x^8 - 11561*x^7 + 79270*x^6 - 49005*x^5 + 81626*x^4 - 70530*x^3 + 35205*x^2 - 15236*x + 3229, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3229, -15236, 35205, -70530, 81626, -49005, 79270, -11561, -40976, 16909, -9040, 37520, -26180, -4171, 11604, -15748, 6967, 4025, -2152, 1331, -120, -225, -18, 7, 5, -10, 4, -2, 1]);
 

\( x^{28} - 2 x^{27} + 4 x^{26} - 10 x^{25} + 5 x^{24} + 7 x^{23} - 18 x^{22} - 225 x^{21} - 120 x^{20} + 1331 x^{19} - 2152 x^{18} + 4025 x^{17} + 6967 x^{16} - 15748 x^{15} + 11604 x^{14} - 4171 x^{13} - 26180 x^{12} + 37520 x^{11} - 9040 x^{10} + 16909 x^{9} - 40976 x^{8} - 11561 x^{7} + 79270 x^{6} - 49005 x^{5} + 81626 x^{4} - 70530 x^{3} + 35205 x^{2} - 15236 x + 3229 \)

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(650754790224967946962108515803180182241777\)\(\medspace = 7^{21}\cdot 71^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $31.14$
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:  $7, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{11} a^{24} - \frac{2}{11} a^{23} - \frac{4}{11} a^{21} - \frac{5}{11} a^{20} - \frac{4}{11} a^{19} - \frac{1}{11} a^{18} - \frac{5}{11} a^{16} + \frac{3}{11} a^{15} + \frac{5}{11} a^{14} - \frac{3}{11} a^{13} + \frac{4}{11} a^{12} - \frac{3}{11} a^{11} - \frac{4}{11} a^{10} - \frac{1}{11} a^{8} + \frac{5}{11} a^{7} - \frac{4}{11} a^{6} - \frac{5}{11} a^{5} - \frac{3}{11} a^{4} + \frac{2}{11} a^{3} + \frac{5}{11} a^{2} + \frac{2}{11} a + \frac{2}{11}$, $\frac{1}{451} a^{25} - \frac{17}{451} a^{24} + \frac{96}{451} a^{23} - \frac{224}{451} a^{22} + \frac{15}{41} a^{21} - \frac{50}{451} a^{20} + \frac{158}{451} a^{19} - \frac{172}{451} a^{18} - \frac{2}{11} a^{17} + \frac{56}{451} a^{16} - \frac{62}{451} a^{15} - \frac{155}{451} a^{14} + \frac{159}{451} a^{13} + \frac{135}{451} a^{12} - \frac{190}{451} a^{11} + \frac{225}{451} a^{10} + \frac{54}{451} a^{9} - \frac{57}{451} a^{8} + \frac{42}{451} a^{7} - \frac{12}{41} a^{6} + \frac{116}{451} a^{5} + \frac{69}{451} a^{4} + \frac{129}{451} a^{3} - \frac{40}{451} a^{2} + \frac{104}{451} a + \frac{190}{451}$, $\frac{1}{134849} a^{26} - \frac{9}{10373} a^{25} + \frac{5117}{134849} a^{24} + \frac{49380}{134849} a^{23} + \frac{57743}{134849} a^{22} + \frac{2154}{5863} a^{21} + \frac{54399}{134849} a^{20} + \frac{33884}{134849} a^{19} - \frac{8}{5863} a^{18} - \frac{313}{134849} a^{17} - \frac{65563}{134849} a^{16} - \frac{7895}{134849} a^{15} - \frac{10581}{134849} a^{14} + \frac{20274}{134849} a^{13} + \frac{3438}{10373} a^{12} - \frac{229}{533} a^{11} - \frac{49260}{134849} a^{10} + \frac{65350}{134849} a^{9} + \frac{60600}{134849} a^{8} + \frac{48353}{134849} a^{7} + \frac{16719}{134849} a^{6} + \frac{53946}{134849} a^{5} - \frac{64540}{134849} a^{4} + \frac{41508}{134849} a^{3} + \frac{49573}{134849} a^{2} - \frac{20255}{134849} a + \frac{59802}{134849}$, $\frac{1}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{27} - \frac{113134905566530706811559113312678818587548214442522603782904566157}{39260661759346726316812781188894720845193079248357356040930365699397529} a^{26} + \frac{19356916726025190826086381961120122212573950843397707826384720083048}{18776838232731043021084373612080083882483646597040474628271044464929253} a^{25} + \frac{1312761980235379843710061754532302858908812647212891464099391018659347}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{24} - \frac{136782935238788093970090396682275180094960094601348474340393881431452898}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{23} - \frac{166479922894052358520050799981304199572012870271073927244923896728546400}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{22} - \frac{156789758199688487383793107137118289434625316505382387920195378159387558}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{21} + \frac{201418185104024743440548182833015494366843397211995947486655925138034746}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{20} + \frac{3168894623514218535029144920780311129509456122068857052000765989673967}{25403957608989058204996505475167172311595521866584171555896118981963107} a^{19} - \frac{197616535936250908029983124748946075668338353274303537349470059756847969}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{18} + \frac{205123938157650579232047662919974574018512826393333913944743835004783184}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{17} - \frac{10763230288960023177638660799261246191461262016857007383691703420574891}{33220559950216460729610814852141686869009528594763916650018001745644063} a^{16} + \frac{214967453135729111445933216962900390289523948457333774793791472602573921}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{15} + \frac{41705272013482774998176634421226959135332626603622248238867508047661018}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{14} + \frac{200164968824644226357271738987020222551306934945589891028133462304466911}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{13} - \frac{139627541395966865255992107048652296975383258175288992246857186118802846}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{12} - \frac{157585217210292383895713623616011594561950659946150538600319322663477690}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{11} - \frac{92075553650297994457772865471686211669872712540813450994477881971003186}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{10} - \frac{3958599902865413109826946442706030571666008721158145570722130194187644}{39260661759346726316812781188894720845193079248357356040930365699397529} a^{9} + \frac{11666067628794916188902006214646486654267448666363490286081128099126233}{39260661759346726316812781188894720845193079248357356040930365699397529} a^{8} + \frac{7611935998483498827904910109864903358089636857823827292101018392755315}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{7} - \frac{969116201981270831275712708789046028033000130393819455275258514758633}{18776838232731043021084373612080083882483646597040474628271044464929253} a^{6} + \frac{126593530876336971759058205223842941811177112152058010531143343410165247}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{5} + \frac{149477583616998926481732847166129502511353646613701664636036354760833237}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{4} - \frac{182740573406511917401289303161894846881581453820675624001390077499312350}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{3} + \frac{104936970092210821020883062315924594845647218926132817533436126738247158}{431867279352813989484940593077841929297123871731930916450234022693372819} a^{2} - \frac{9175057651028908598318277728106428313252800168787012732330302656159310}{18776838232731043021084373612080083882483646597040474628271044464929253} a + \frac{189375209722564092760231321541384981859979253100976062062127787441559932}{431867279352813989484940593077841929297123871731930916450234022693372819}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $13$
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:  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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 3185481783.8925004 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 3185481783.8925004 \cdot 1}{2\sqrt{650754790224967946962108515803180182241777}}\approx 0.295090073420219$ (assuming GRH)

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

\(\Q(\sqrt{-7}) \), 4.0.24353.1, 7.1.357911.1, 14.0.105496092121152103.1

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

Sibling fields

Degree 28 sibling: 28.2.46203590105972724234309704622025792939166167.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{4}$ $28$ $28$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ $28$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
$7$7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$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$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.497.2t1.a.a$1$ $ 7 \cdot 71 $ \(\Q(\sqrt{497}) \) $C_2$ (as 2T1) $1$ $1$
1.71.2t1.a.a$1$ $ 71 $ \(\Q(\sqrt{-71}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.7.2t1.a.a$1$ $ 7 $ \(\Q(\sqrt{-7}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3479.4t3.c.a$2$ $ 7^{2} \cdot 71 $ 4.2.1729063.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.3479.14t3.a.c$2$ $ 7^{2} \cdot 71 $ 14.2.7490222540601799313.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.3479.14t3.a.a$2$ $ 7^{2} \cdot 71 $ 14.2.7490222540601799313.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.71.7t2.a.b$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.71.7t2.a.a$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3479.14t3.a.b$2$ $ 7^{2} \cdot 71 $ 14.2.7490222540601799313.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.71.7t2.a.c$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3479.28t10.b.f$2$ $ 7^{2} \cdot 71 $ 28.0.650754790224967946962108515803180182241777.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3479.28t10.b.e$2$ $ 7^{2} \cdot 71 $ 28.0.650754790224967946962108515803180182241777.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3479.28t10.b.d$2$ $ 7^{2} \cdot 71 $ 28.0.650754790224967946962108515803180182241777.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3479.28t10.b.a$2$ $ 7^{2} \cdot 71 $ 28.0.650754790224967946962108515803180182241777.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3479.28t10.b.c$2$ $ 7^{2} \cdot 71 $ 28.0.650754790224967946962108515803180182241777.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3479.28t10.b.b$2$ $ 7^{2} \cdot 71 $ 28.0.650754790224967946962108515803180182241777.1 $D_{28}$ (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 *.