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Properties

Label 28.0.22791052858...8064.1
Degree $28$
Signature $[0, 14]$
Discriminant $2^{28}\cdot 7^{14}\cdot 29^{24}$
Root discriminant $94.86$
Ramified primes $2, 7, 29$
Class number $3136$ (GRH)
Class group $[4, 28, 28]$ (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![495997441, 0, -748889507, 0, 707029168, 0, -400934959, 0, 167974631, 0, -43934230, 0, 8840544, 0, -578821, 0, 178104, 0, 6532, 0, 14430, 0, 2977, 0, 411, 0, 27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 27*x^26 + 411*x^24 + 2977*x^22 + 14430*x^20 + 6532*x^18 + 178104*x^16 - 578821*x^14 + 8840544*x^12 - 43934230*x^10 + 167974631*x^8 - 400934959*x^6 + 707029168*x^4 - 748889507*x^2 + 495997441)
 
gp: K = bnfinit(x^28 + 27*x^26 + 411*x^24 + 2977*x^22 + 14430*x^20 + 6532*x^18 + 178104*x^16 - 578821*x^14 + 8840544*x^12 - 43934230*x^10 + 167974631*x^8 - 400934959*x^6 + 707029168*x^4 - 748889507*x^2 + 495997441, 1)
 

Normalized defining polynomial

\( x^{28} + 27 x^{26} + 411 x^{24} + 2977 x^{22} + 14430 x^{20} + 6532 x^{18} + 178104 x^{16} - 578821 x^{14} + 8840544 x^{12} - 43934230 x^{10} + 167974631 x^{8} - 400934959 x^{6} + 707029168 x^{4} - 748889507 x^{2} + 495997441 \)

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

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 14]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(22791052858328956460332445607300922515156603344533848064=2^{28}\cdot 7^{14}\cdot 29^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.86$
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:  $2, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(812=2^{2}\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{812}(1,·)$, $\chi_{812}(603,·)$, $\chi_{812}(587,·)$, $\chi_{812}(517,·)$, $\chi_{812}(393,·)$, $\chi_{812}(139,·)$, $\chi_{812}(239,·)$, $\chi_{812}(141,·)$, $\chi_{812}(335,·)$, $\chi_{812}(83,·)$, $\chi_{812}(407,·)$, $\chi_{812}(281,·)$, $\chi_{812}(687,·)$, $\chi_{812}(349,·)$, $\chi_{812}(645,·)$, $\chi_{812}(545,·)$, $\chi_{812}(197,·)$, $\chi_{812}(547,·)$, $\chi_{812}(741,·)$, $\chi_{812}(169,·)$, $\chi_{812}(223,·)$, $\chi_{812}(111,·)$, $\chi_{812}(755,·)$, $\chi_{812}(181,·)$, $\chi_{812}(489,·)$, $\chi_{812}(799,·)$, $\chi_{812}(629,·)$, $\chi_{812}(575,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{17} a^{22} + \frac{7}{17} a^{20} + \frac{6}{17} a^{18} - \frac{8}{17} a^{16} - \frac{3}{17} a^{14} - \frac{4}{17} a^{10} - \frac{6}{17} a^{8} + \frac{7}{17} a^{6} + \frac{8}{17} a^{4} - \frac{7}{17} a^{2} + \frac{5}{17}$, $\frac{1}{17} a^{23} + \frac{7}{17} a^{21} + \frac{6}{17} a^{19} - \frac{8}{17} a^{17} - \frac{3}{17} a^{15} - \frac{4}{17} a^{11} - \frac{6}{17} a^{9} + \frac{7}{17} a^{7} + \frac{8}{17} a^{5} - \frac{7}{17} a^{3} + \frac{5}{17} a$, $\frac{1}{5899} a^{24} + \frac{1}{5899} a^{22} - \frac{1362}{5899} a^{20} + \frac{1248}{5899} a^{18} + \frac{1796}{5899} a^{16} - \frac{2396}{5899} a^{14} - \frac{803}{5899} a^{12} - \frac{16}{5899} a^{10} - \frac{2762}{5899} a^{8} + \frac{59}{347} a^{6} - \frac{2044}{5899} a^{4} + \frac{2189}{5899} a^{2} + \frac{2673}{5899}$, $\frac{1}{5899} a^{25} + \frac{1}{5899} a^{23} - \frac{1362}{5899} a^{21} + \frac{1248}{5899} a^{19} + \frac{1796}{5899} a^{17} - \frac{2396}{5899} a^{15} - \frac{803}{5899} a^{13} - \frac{16}{5899} a^{11} - \frac{2762}{5899} a^{9} + \frac{59}{347} a^{7} - \frac{2044}{5899} a^{5} + \frac{2189}{5899} a^{3} + \frac{2673}{5899} a$, $\frac{1}{3874074053765391278251991125415695934466084294809323} a^{26} - \frac{240510165087645602369902184768004760342799850337}{3874074053765391278251991125415695934466084294809323} a^{24} - \frac{101524877353725671947336651315593424167397726003762}{3874074053765391278251991125415695934466084294809323} a^{22} + \frac{1396648126321254773033681190197882911807603254342988}{3874074053765391278251991125415695934466084294809323} a^{20} + \frac{342318949716324631746705582670624827233260523544960}{3874074053765391278251991125415695934466084294809323} a^{18} - \frac{21593865028744757095410433402456286645272970483226}{3874074053765391278251991125415695934466084294809323} a^{16} - \frac{91153746658026854252640311201674376458738319048130}{3874074053765391278251991125415695934466084294809323} a^{14} + \frac{39197460678361332787342934547019356197377858257037}{94489611067448567762243685985748681328441080361203} a^{12} + \frac{480413606472643642049437459363385884314026470031722}{3874074053765391278251991125415695934466084294809323} a^{10} - \frac{1855960923050008431661952724402139055689737360049324}{3874074053765391278251991125415695934466084294809323} a^{8} - \frac{1928123153335777057672101734276144381113842803963353}{3874074053765391278251991125415695934466084294809323} a^{6} + \frac{1281917996560290577331722296794632443990011616761377}{3874074053765391278251991125415695934466084294809323} a^{4} + \frac{767597288149998548487017302012640419407639727983875}{3874074053765391278251991125415695934466084294809323} a^{2} - \frac{772846648302245651413468808592702734664272495472077}{3874074053765391278251991125415695934466084294809323}$, $\frac{1}{86279503251409029157950094354132964156494163329698432533} a^{27} + \frac{148485006364092738073175651238096075227170765263603}{5075264897141707597526476138478409656264362548805790149} a^{25} + \frac{338555856582409887663351190497690425702835691309430851}{86279503251409029157950094354132964156494163329698432533} a^{23} - \frac{34211229726930729552219314069551533276097512091387079666}{86279503251409029157950094354132964156494163329698432533} a^{21} - \frac{2623065176417410928406042165979615390363214426935563619}{86279503251409029157950094354132964156494163329698432533} a^{19} - \frac{19164212183601040349505205325043517746257713675970734007}{86279503251409029157950094354132964156494163329698432533} a^{17} + \frac{26775131328684501752633160432551916592226811919413975706}{86279503251409029157950094354132964156494163329698432533} a^{15} + \frac{500809228278655875147561013100156268030522876974877102}{2104378128083147052632929130588608881865711300724352013} a^{13} + \frac{36058482789989668924292193284327251700587539429948421009}{86279503251409029157950094354132964156494163329698432533} a^{11} - \frac{31054134350060989533573099637089324538264967781029608015}{86279503251409029157950094354132964156494163329698432533} a^{9} + \frac{24164327788536607444224547968842156333833366141804341157}{86279503251409029157950094354132964156494163329698432533} a^{7} - \frac{7998007193912315080526593853552242747015935965541084216}{86279503251409029157950094354132964156494163329698432533} a^{5} - \frac{9626004328664205510652583633896135642111104696277894360}{86279503251409029157950094354132964156494163329698432533} a^{3} - \frac{29158424779211066294061660995262597938585446836261673974}{86279503251409029157950094354132964156494163329698432533} a$

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

Class group and class number

$C_{4}\times C_{28}\times C_{28}$, which has order $3136$ (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:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{9782543304320248700251019}{205754832990906140078931100065719} a^{27} + \frac{253251841336019702913259789}{205754832990906140078931100065719} a^{25} + \frac{3666110652950461555928384359}{205754832990906140078931100065719} a^{23} + \frac{22888203162067946290263959375}{205754832990906140078931100065719} a^{21} + \frac{80351826153513962225899677975}{205754832990906140078931100065719} a^{19} - \frac{328085531477014816014206006597}{205754832990906140078931100065719} a^{17} + \frac{370908401486471590791555243999}{205754832990906140078931100065719} a^{15} - \frac{248338140287678230624928130921}{5018410560753808294608075611359} a^{13} + \frac{79300830764757660494371007358738}{205754832990906140078931100065719} a^{11} - \frac{509985551318620461101034274888946}{205754832990906140078931100065719} a^{9} + \frac{1651613258390634666509383141233011}{205754832990906140078931100065719} a^{7} - \frac{3989771569608108158499894023017374}{205754832990906140078931100065719} a^{5} + \frac{5352485116038112886519903841895783}{205754832990906140078931100065719} a^{3} - \frac{4735567434161807016480643192805695}{205754832990906140078931100065719} a \) (order $4$)
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:  \( 4951230435467.248 \) (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

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{7}) \), \(\Q(i, \sqrt{7})\), 7.7.594823321.1, 14.0.291381688005381590432263.1, 14.0.5796901408038404767744.1, 14.14.4773997576280171977642196992.1

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 ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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

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
$2$2.14.14.38$x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.38$x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
7Data not computed
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$