Properties

Label 28.28.642...625.1
Degree $28$
Signature $[28, 0]$
Discriminant $6.426\times 10^{47}$
Root discriminant $50.98$
Ramified primes $5, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - 40*x^26 + 35*x^25 + 667*x^24 - 497*x^23 - 6073*x^22 + 3733*x^21 + 33381*x^20 - 16356*x^19 - 116335*x^18 + 43955*x^17 + 264151*x^16 - 74631*x^15 - 396001*x^14 + 80966*x^13 + 391804*x^12 - 55684*x^11 - 251669*x^10 + 23519*x^9 + 101079*x^8 - 5634*x^7 - 23674*x^6 + 574*x^5 + 2842*x^4 + 28*x^3 - 133*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^28 - x^27 - 40*x^26 + 35*x^25 + 667*x^24 - 497*x^23 - 6073*x^22 + 3733*x^21 + 33381*x^20 - 16356*x^19 - 116335*x^18 + 43955*x^17 + 264151*x^16 - 74631*x^15 - 396001*x^14 + 80966*x^13 + 391804*x^12 - 55684*x^11 - 251669*x^10 + 23519*x^9 + 101079*x^8 - 5634*x^7 - 23674*x^6 + 574*x^5 + 2842*x^4 + 28*x^3 - 133*x^2 - 7*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, -133, 28, 2842, 574, -23674, -5634, 101079, 23519, -251669, -55684, 391804, 80966, -396001, -74631, 264151, 43955, -116335, -16356, 33381, 3733, -6073, -497, 667, 35, -40, -1, 1]);
 

\( x^{28} - x^{27} - 40 x^{26} + 35 x^{25} + 667 x^{24} - 497 x^{23} - 6073 x^{22} + 3733 x^{21} + 33381 x^{20} - 16356 x^{19} - 116335 x^{18} + 43955 x^{17} + 264151 x^{16} - 74631 x^{15} - 396001 x^{14} + 80966 x^{13} + 391804 x^{12} - 55684 x^{11} - 251669 x^{10} + 23519 x^{9} + 101079 x^{8} - 5634 x^{7} - 23674 x^{6} + 574 x^{5} + 2842 x^{4} + 28 x^{3} - 133 x^{2} - 7 x + 1 \)

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[28, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(642581186433047492874742549394260203375244140625\)\(\medspace = 5^{14}\cdot 29^{26}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $50.98$
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:  $5, 29$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $28$
This field is Galois and abelian over $\Q$.
Conductor:  \(145=5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{145}(64,·)$, $\chi_{145}(1,·)$, $\chi_{145}(4,·)$, $\chi_{145}(6,·)$, $\chi_{145}(129,·)$, $\chi_{145}(136,·)$, $\chi_{145}(9,·)$, $\chi_{145}(74,·)$, $\chi_{145}(139,·)$, $\chi_{145}(141,·)$, $\chi_{145}(16,·)$, $\chi_{145}(81,·)$, $\chi_{145}(86,·)$, $\chi_{145}(24,·)$, $\chi_{145}(91,·)$, $\chi_{145}(94,·)$, $\chi_{145}(96,·)$, $\chi_{145}(144,·)$, $\chi_{145}(34,·)$, $\chi_{145}(36,·)$, $\chi_{145}(71,·)$, $\chi_{145}(109,·)$, $\chi_{145}(111,·)$, $\chi_{145}(49,·)$, $\chi_{145}(51,·)$, $\chi_{145}(54,·)$, $\chi_{145}(121,·)$, $\chi_{145}(59,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{59} a^{25} - \frac{25}{59} a^{24} + \frac{28}{59} a^{22} + \frac{5}{59} a^{21} - \frac{13}{59} a^{20} - \frac{29}{59} a^{19} + \frac{25}{59} a^{18} + \frac{9}{59} a^{17} - \frac{22}{59} a^{16} - \frac{25}{59} a^{15} + \frac{19}{59} a^{14} - \frac{21}{59} a^{13} + \frac{26}{59} a^{12} + \frac{8}{59} a^{11} - \frac{11}{59} a^{10} - \frac{17}{59} a^{9} + \frac{16}{59} a^{8} - \frac{15}{59} a^{7} - \frac{13}{59} a^{6} + \frac{21}{59} a^{5} + \frac{27}{59} a^{4} - \frac{16}{59} a^{3} + \frac{25}{59} a^{2} - \frac{7}{59} a - \frac{27}{59}$, $\frac{1}{59} a^{26} + \frac{24}{59} a^{24} + \frac{28}{59} a^{23} - \frac{3}{59} a^{22} - \frac{6}{59} a^{21} + \frac{8}{59} a^{19} - \frac{15}{59} a^{18} + \frac{26}{59} a^{17} + \frac{15}{59} a^{16} - \frac{16}{59} a^{15} - \frac{18}{59} a^{14} - \frac{27}{59} a^{13} + \frac{9}{59} a^{12} + \frac{12}{59} a^{11} + \frac{3}{59} a^{10} + \frac{4}{59} a^{9} - \frac{28}{59} a^{8} + \frac{25}{59} a^{7} - \frac{9}{59} a^{6} + \frac{21}{59} a^{5} + \frac{10}{59} a^{4} - \frac{21}{59} a^{3} + \frac{28}{59} a^{2} - \frac{25}{59} a - \frac{26}{59}$, $\frac{1}{1174451931043404464510641028789} a^{27} + \frac{4664799318915561977227153332}{1174451931043404464510641028789} a^{26} - \frac{110809578378861470963539376}{69085407708435556735920060517} a^{25} - \frac{164622800266790736038061199004}{1174451931043404464510641028789} a^{24} + \frac{181964440414601394490988721586}{1174451931043404464510641028789} a^{23} - \frac{12744189033319699827018283214}{69085407708435556735920060517} a^{22} - \frac{198130269341417057305802908344}{1174451931043404464510641028789} a^{21} + \frac{219739351239271133761406432576}{1174451931043404464510641028789} a^{20} + \frac{40384319278439428721390253374}{1174451931043404464510641028789} a^{19} + \frac{147324928552331529180147580278}{1174451931043404464510641028789} a^{18} - \frac{42029580497118862738136615318}{1174451931043404464510641028789} a^{17} - \frac{389751348565644730293997862490}{1174451931043404464510641028789} a^{16} + \frac{434768975132253161969213522229}{1174451931043404464510641028789} a^{15} + \frac{248394608026106213821969851080}{1174451931043404464510641028789} a^{14} + \frac{570527313822837965254608767218}{1174451931043404464510641028789} a^{13} + \frac{108694819029305994132019389541}{1174451931043404464510641028789} a^{12} + \frac{435324265311585056536518735532}{1174451931043404464510641028789} a^{11} - \frac{457221319928289594963530495368}{1174451931043404464510641028789} a^{10} - \frac{384178468361149903985654966053}{1174451931043404464510641028789} a^{9} + \frac{22609238791726581558521141620}{1174451931043404464510641028789} a^{8} + \frac{361244437823249756613427190069}{1174451931043404464510641028789} a^{7} - \frac{553987219776826163652268219043}{1174451931043404464510641028789} a^{6} - \frac{32995348050508953620351723898}{69085407708435556735920060517} a^{5} + \frac{238800709684541288010580622570}{1174451931043404464510641028789} a^{4} - \frac{18379768169071649419415410836}{1174451931043404464510641028789} a^{3} - \frac{447048766234291197554777418226}{1174451931043404464510641028789} a^{2} - \frac{82008374321897316340359420963}{1174451931043404464510641028789} a + \frac{446285954461926414459632231309}{1174451931043404464510641028789}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $27$
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:  \( 512915496404520.1 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{28}\cdot(2\pi)^{0}\cdot 512915496404520.1 \cdot 2}{2\sqrt{642581186433047492874742549394260203375244140625}}\approx 0.171759867298331$ (assuming GRH)

Galois group

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

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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 7.7.594823321.1, 14.14.27641779937927268828125.1, \(\Q(\zeta_{29})^+\), 14.14.801611618199890796015625.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

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

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
$5$5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$