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Properties

Label 28.28.1351715799...4944.1
Degree $28$
Signature $[28, 0]$
Discriminant $2^{28}\cdot 3^{14}\cdot 29^{26}$
Root discriminant $78.98$
Ramified primes $2, 3, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 56, -67, -7818, 15255, 260140, -481394, -3052488, 4786431, 12642698, -15790909, -22027400, 22549873, 19086334, -16632140, -9204326, 7045255, 2646496, -1822856, -470232, 296883, 51962, -30502, -3470, 1915, 128, -67, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 2*x^27 - 67*x^26 + 128*x^25 + 1915*x^24 - 3470*x^23 - 30502*x^22 + 51962*x^21 + 296883*x^20 - 470232*x^19 - 1822856*x^18 + 2646496*x^17 + 7045255*x^16 - 9204326*x^15 - 16632140*x^14 + 19086334*x^13 + 22549873*x^12 - 22027400*x^11 - 15790909*x^10 + 12642698*x^9 + 4786431*x^8 - 3052488*x^7 - 481394*x^6 + 260140*x^5 + 15255*x^4 - 7818*x^3 - 67*x^2 + 56*x + 1)
 
gp: K = bnfinit(x^28 - 2*x^27 - 67*x^26 + 128*x^25 + 1915*x^24 - 3470*x^23 - 30502*x^22 + 51962*x^21 + 296883*x^20 - 470232*x^19 - 1822856*x^18 + 2646496*x^17 + 7045255*x^16 - 9204326*x^15 - 16632140*x^14 + 19086334*x^13 + 22549873*x^12 - 22027400*x^11 - 15790909*x^10 + 12642698*x^9 + 4786431*x^8 - 3052488*x^7 - 481394*x^6 + 260140*x^5 + 15255*x^4 - 7818*x^3 - 67*x^2 + 56*x + 1, 1)
 

Normalized defining polynomial

\( x^{28} - 2 x^{27} - 67 x^{26} + 128 x^{25} + 1915 x^{24} - 3470 x^{23} - 30502 x^{22} + 51962 x^{21} + 296883 x^{20} - 470232 x^{19} - 1822856 x^{18} + 2646496 x^{17} + 7045255 x^{16} - 9204326 x^{15} - 16632140 x^{14} + 19086334 x^{13} + 22549873 x^{12} - 22027400 x^{11} - 15790909 x^{10} + 12642698 x^{9} + 4786431 x^{8} - 3052488 x^{7} - 481394 x^{6} + 260140 x^{5} + 15255 x^{4} - 7818 x^{3} - 67 x^{2} + 56 x + 1 \)

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

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[28, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(135171579942192030712001098144632895138136441638354944=2^{28}\cdot 3^{14}\cdot 29^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.98$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(348=2^{2}\cdot 3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{348}(1,·)$, $\chi_{348}(323,·)$, $\chi_{348}(227,·)$, $\chi_{348}(325,·)$, $\chi_{348}(71,·)$, $\chi_{348}(265,·)$, $\chi_{348}(241,·)$, $\chi_{348}(13,·)$, $\chi_{348}(335,·)$, $\chi_{348}(83,·)$, $\chi_{348}(277,·)$, $\chi_{348}(23,·)$, $\chi_{348}(25,·)$, $\chi_{348}(347,·)$, $\chi_{348}(107,·)$, $\chi_{348}(289,·)$, $\chi_{348}(35,·)$, $\chi_{348}(167,·)$, $\chi_{348}(169,·)$, $\chi_{348}(299,·)$, $\chi_{348}(109,·)$, $\chi_{348}(239,·)$, $\chi_{348}(49,·)$, $\chi_{348}(179,·)$, $\chi_{348}(181,·)$, $\chi_{348}(121,·)$, $\chi_{348}(313,·)$, $\chi_{348}(59,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{59} a^{25} - \frac{28}{59} a^{24} + \frac{8}{59} a^{23} - \frac{9}{59} a^{22} + \frac{20}{59} a^{21} + \frac{25}{59} a^{20} - \frac{6}{59} a^{19} - \frac{14}{59} a^{18} + \frac{7}{59} a^{17} + \frac{23}{59} a^{15} - \frac{8}{59} a^{14} + \frac{4}{59} a^{13} + \frac{16}{59} a^{12} - \frac{26}{59} a^{11} - \frac{13}{59} a^{10} - \frac{23}{59} a^{9} + \frac{19}{59} a^{8} - \frac{18}{59} a^{7} - \frac{21}{59} a^{6} + \frac{13}{59} a^{5} + \frac{7}{59} a^{4} + \frac{24}{59} a^{3} + \frac{4}{59} a^{2} + \frac{18}{59} a - \frac{23}{59}$, $\frac{1}{59} a^{26} - \frac{9}{59} a^{24} - \frac{21}{59} a^{23} + \frac{4}{59} a^{22} - \frac{5}{59} a^{21} - \frac{14}{59} a^{20} - \frac{5}{59} a^{19} + \frac{28}{59} a^{18} + \frac{19}{59} a^{17} + \frac{23}{59} a^{16} - \frac{13}{59} a^{15} + \frac{16}{59} a^{14} + \frac{10}{59} a^{13} + \frac{9}{59} a^{12} + \frac{26}{59} a^{11} + \frac{26}{59} a^{10} + \frac{24}{59} a^{9} - \frac{17}{59} a^{8} + \frac{6}{59} a^{7} + \frac{15}{59} a^{6} + \frac{17}{59} a^{5} - \frac{16}{59} a^{4} + \frac{27}{59} a^{3} + \frac{12}{59} a^{2} + \frac{9}{59} a + \frac{5}{59}$, $\frac{1}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{27} - \frac{9183407243623464517650231709450524925202101371996875605186354030363}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{26} - \frac{195880563754888592523937369869671912657286468001334848193811615122407}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{25} - \frac{7634036178310197274710263889281321634940411279688711247328650704215361}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{24} + \frac{4563594055668292409095766192922410165773002660040580708640323571164881}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{23} - \frac{10406220734721690917614302579881477739151668661304898706004452148339312}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{22} - \frac{5954999660841970472950179445136030709728725328012763149596695120414596}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{21} + \frac{4342659008983754638410783473913826237046508719477374102804765698608828}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{20} - \frac{8033318575676300138058403706565270149341910289070523016386138913899000}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{19} - \frac{10245424881955013443433928112613720715853587366102380496737737377745203}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{18} + \frac{2497157168228148789794175243014132342757862348652801226945337278830552}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{17} + \frac{9438512473365773109807874125928651305035569372837972781519471463473196}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{16} + \frac{155213998932922828086217214647059084270677932658810845013634052763840}{602819703467043848060421890711681151311392561343617383542891921702627} a^{15} - \frac{140368917726243004831970915711460431004979501827919204420289955998712}{418908607494047419838598263036930969555374491781157842800992691352673} a^{14} + \frac{4497119497785264998402141637046936678991325396066501916021128764042906}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{13} - \frac{3198464781631911330688715742264258001869167262706372056114747502223000}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{12} + \frac{5496571902735466352496100446631380348486677458055198653788551338189756}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{11} - \frac{2124333419470539401940133252461760091120305718062260430379501886192616}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{10} + \frac{1484606763366516203974288947560296384932279840824213062974960133681871}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{9} - \frac{2137528965141030734393371023365508372598008061689583670397618293073670}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{8} - \frac{5975806751722723493333498150117978753596175703807108415769602225342954}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{7} - \frac{2017826075936109774941441880427412701884453438489241107023628788412288}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{6} + \frac{4233397537486293133038093129664072420046179526950493292938647839291820}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{5} + \frac{2315813288890254736961975419281358395163991235706453730329031104716014}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{4} + \frac{115792572754090150550730801133481544723241539992178577879735773863128}{418908607494047419838598263036930969555374491781157842800992691352673} a^{3} + \frac{4014089541427069965271046800890481294839600660790135192689600795024825}{24715607842148797770477297519178927203767095015088312725258568789807707} a^{2} + \frac{6945109601906773901662527199712329404743482609748886017502325200142086}{24715607842148797770477297519178927203767095015088312725258568789807707} a + \frac{8541638716734415205829051616116469058024631450845387761284674667131134}{24715607842148797770477297519178927203767095015088312725258568789807707}$

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

Class group and class number

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

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $27$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 442900936905773100 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{14}$ (as 28T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{87}) \), \(\Q(\sqrt{3}, \sqrt{29})\), 7.7.594823321.1, 14.14.12677823379379991227056128.1, \(\Q(\zeta_{29})^+\), 14.14.367656878002019745584627712.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
3Data not computed
29Data not computed