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Properties

Label 20.0.57260615372...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{20}\cdot 7^{8}\cdot 97$
Root discriminant $38.72$
Ramified primes $2, 5, 7, 97$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group 20T633

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![185, -470, 860, 7680, -21025, -8876, 90390, -140080, 103320, -28940, -8546, 7060, -20, -880, -170, 188, 35, -10, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 - 10*x^17 + 35*x^16 + 188*x^15 - 170*x^14 - 880*x^13 - 20*x^12 + 7060*x^11 - 8546*x^10 - 28940*x^9 + 103320*x^8 - 140080*x^7 + 90390*x^6 - 8876*x^5 - 21025*x^4 + 7680*x^3 + 860*x^2 - 470*x + 185)
 
gp: K = bnfinit(x^20 - 10*x^18 - 10*x^17 + 35*x^16 + 188*x^15 - 170*x^14 - 880*x^13 - 20*x^12 + 7060*x^11 - 8546*x^10 - 28940*x^9 + 103320*x^8 - 140080*x^7 + 90390*x^6 - 8876*x^5 - 21025*x^4 + 7680*x^3 + 860*x^2 - 470*x + 185, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} - 10 x^{17} + 35 x^{16} + 188 x^{15} - 170 x^{14} - 880 x^{13} - 20 x^{12} + 7060 x^{11} - 8546 x^{10} - 28940 x^{9} + 103320 x^{8} - 140080 x^{7} + 90390 x^{6} - 8876 x^{5} - 21025 x^{4} + 7680 x^{3} + 860 x^{2} - 470 x + 185 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(57260615372800000000000000000000=2^{30}\cdot 5^{20}\cdot 7^{8}\cdot 97\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.72$
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:  $2, 5, 7, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{16} - \frac{1}{2} a^{9} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{878427381194285661776904517346037570056644} a^{19} - \frac{43605063650020760734282607789690249103113}{878427381194285661776904517346037570056644} a^{18} + \frac{127470653980638225437943234833857902727685}{878427381194285661776904517346037570056644} a^{17} + \frac{83848841969338184257388669532386832842847}{878427381194285661776904517346037570056644} a^{16} + \frac{47020451434412274697503187008817520484125}{439213690597142830888452258673018785028322} a^{15} - \frac{2631287195180627237897532287256750182958}{219606845298571415444226129336509392514161} a^{14} + \frac{107291323627871267786317602341554909291835}{439213690597142830888452258673018785028322} a^{13} + \frac{7647611530775442340060728863249365903563}{219606845298571415444226129336509392514161} a^{12} - \frac{39465975508068469806731673510132108668823}{439213690597142830888452258673018785028322} a^{11} - \frac{953952768179504208562888403393524405663}{14168183567649768738337169634613509194462} a^{10} - \frac{46623563709107687663938748122821025279616}{219606845298571415444226129336509392514161} a^{9} + \frac{93402980844404091418445834565426553832574}{219606845298571415444226129336509392514161} a^{8} + \frac{14212556075337387002148501718892421355479}{439213690597142830888452258673018785028322} a^{7} + \frac{107065811351083090309312667844060940403122}{219606845298571415444226129336509392514161} a^{6} + \frac{142859710731462937045625882661513365152995}{439213690597142830888452258673018785028322} a^{5} + \frac{105939289808374641221296920035814655963342}{219606845298571415444226129336509392514161} a^{4} + \frac{20319009469029347202744800137364867261861}{878427381194285661776904517346037570056644} a^{3} - \frac{275163434137064101774851204131769774697489}{878427381194285661776904517346037570056644} a^{2} - \frac{407777341877258029592092976438803559869229}{878427381194285661776904517346037570056644} a - \frac{424215086695617966630177626949706867315843}{878427381194285661776904517346037570056644}$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1041095041344421983030}{441857951046636782012681} a^{19} - \frac{176320273523476946210}{441857951046636782012681} a^{18} + \frac{10354857297172686314345}{441857951046636782012681} a^{17} + \frac{24739979172393118961445}{883715902093273564025362} a^{16} - \frac{67259171858915869757609}{883715902093273564025362} a^{15} - \frac{202318018805949999816460}{441857951046636782012681} a^{14} + \frac{136878257830242723308810}{441857951046636782012681} a^{13} + \frac{1862188362682916381321135}{883715902093273564025362} a^{12} + \frac{434320246299823941597665}{883715902093273564025362} a^{11} - \frac{233744634290160728514100}{14253482291826992968151} a^{10} + \frac{7572635982056391264502675}{441857951046636782012681} a^{9} + \frac{61941795940854762069487355}{883715902093273564025362} a^{8} - \frac{202363258180019735616250835}{883715902093273564025362} a^{7} + \frac{129874711390123276307252770}{441857951046636782012681} a^{6} - \frac{80189815189681018004071262}{441857951046636782012681} a^{5} + \frac{16159724966080292342819325}{883715902093273564025362} a^{4} + \frac{27822873026964577536571545}{883715902093273564025362} a^{3} - \frac{3756568418021260549727005}{441857951046636782012681} a^{2} - \frac{931836882507170528552615}{441857951046636782012681} a + \frac{217820381586181337494962}{441857951046636782012681} \) (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:  \( 23992690.3729 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T633:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40960
The 124 conjugacy class representatives for t20n633 are not computed
Character table for t20n633 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.5.2450000.1, 10.0.384160000000000.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ $20$

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
$5$5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.10.10.10$x^{10} + 10 x^{8} + 5 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 2$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97Data not computed