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Properties

Label 16.0.18616441050...9936.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 7^{6}\cdot 193^{4}\cdot 223^{4}$
Root discriminant $438.40$
Ramified primes $2, 7, 193, 223$
Class number $19398312960$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 75774660]$ (GRH)
Galois group $(C_2^2\times C_4).C_2^4$ (as 16T471)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11373046199801545201, -1124404063609193536, 706802171767051960, -34925731551753120, 14342083431686636, -406917584361472, 140107567212424, -2278581253952, 749980009966, -6381521824, 2278069768, -8222112, 3820332, -3680, 3192, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 3192*x^14 - 3680*x^13 + 3820332*x^12 - 8222112*x^11 + 2278069768*x^10 - 6381521824*x^9 + 749980009966*x^8 - 2278581253952*x^7 + 140107567212424*x^6 - 406917584361472*x^5 + 14342083431686636*x^4 - 34925731551753120*x^3 + 706802171767051960*x^2 - 1124404063609193536*x + 11373046199801545201)
 
gp: K = bnfinit(x^16 + 3192*x^14 - 3680*x^13 + 3820332*x^12 - 8222112*x^11 + 2278069768*x^10 - 6381521824*x^9 + 749980009966*x^8 - 2278581253952*x^7 + 140107567212424*x^6 - 406917584361472*x^5 + 14342083431686636*x^4 - 34925731551753120*x^3 + 706802171767051960*x^2 - 1124404063609193536*x + 11373046199801545201, 1)
 

Normalized defining polynomial

\( x^{16} + 3192 x^{14} - 3680 x^{13} + 3820332 x^{12} - 8222112 x^{11} + 2278069768 x^{10} - 6381521824 x^{9} + 749980009966 x^{8} - 2278581253952 x^{7} + 140107567212424 x^{6} - 406917584361472 x^{5} + 14342083431686636 x^{4} - 34925731551753120 x^{3} + 706802171767051960 x^{2} - 1124404063609193536 x + 11373046199801545201 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1861644105089180838863336343231454234279936=2^{62}\cdot 7^{6}\cdot 193^{4}\cdot 223^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $438.40$
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:  $2, 7, 193, 223$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{12} a^{13} - \frac{1}{4} a^{9} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{12} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{5}{12} a - \frac{1}{6}$, $\frac{1}{3301065828} a^{14} + \frac{40348639}{3301065828} a^{13} + \frac{1009097}{91696273} a^{12} + \frac{87611167}{1100355276} a^{11} + \frac{449564}{275088819} a^{10} + \frac{15366503}{550177638} a^{9} + \frac{660095989}{3301065828} a^{8} - \frac{718052}{91696273} a^{7} + \frac{54832941}{366785092} a^{6} - \frac{1557216923}{3301065828} a^{5} + \frac{769822387}{1650532914} a^{4} - \frac{93149609}{366785092} a^{3} - \frac{82074005}{550177638} a^{2} - \frac{173014241}{550177638} a + \frac{1483695865}{3301065828}$, $\frac{1}{27134198842565643939452000759202761126879346846394884185405217702515069239935195258347375259485271507076476272469324} a^{15} - \frac{523551546242127952571897780451742132412919553708516462001598663263987104167618320665195286875578631343722}{6783549710641410984863000189800690281719836711598721046351304425628767309983798814586843814871317876769119068117331} a^{14} - \frac{274694498658465092412666600424837196657300649437236328875579910252882914590580388492673718175746314848019696245840}{6783549710641410984863000189800690281719836711598721046351304425628767309983798814586843814871317876769119068117331} a^{13} - \frac{280737635977389621220956915348297694258203092443015053248930519457566077832722409653552856501754012642311981639851}{4522366473760940656575333459867126854479891141065814030900869617085844873322532543057895876580878584512746045411554} a^{12} + \frac{72267489977995616104328141074668268135151836999633293576548054807404580069488321986626948606837282551427409606359}{1004970327502431257016740768859361523217753586903514229089082137130187749627229454012865750351306352113943565647012} a^{11} - \frac{25280857490141916953977011421690824812457326439152812992008251193171706002398062857052730193401953716800640787283}{1507455491253646885525111153289042284826630380355271343633623205695281624440844181019298625526959528170915348470518} a^{10} - \frac{3135011231410659231567683510281339952392565367248893545459490126781031461150340545050235641193599857853244091303985}{13567099421282821969726000379601380563439673423197442092702608851257534619967597629173687629742635753538238136234662} a^{9} + \frac{2278326366601470727911983866972647992885857525539152688002698516281280690087091348301869994438532933470136285695767}{13567099421282821969726000379601380563439673423197442092702608851257534619967597629173687629742635753538238136234662} a^{8} - \frac{109112149909532292327299687718225108098402558935171019316767663706452021456053085736113149974833674143325097738975}{3014910982507293771050222306578084569653260760710542687267246411390563248881688362038597251053919056341830696941036} a^{7} + \frac{1410648637864190145748190293991880917598265594715208082734605032350881049461149048318246478186132652168050597532671}{13567099421282821969726000379601380563439673423197442092702608851257534619967597629173687629742635753538238136234662} a^{6} + \frac{765005048865364513172108779948115664709398882226046246111758386988456790131862542950374884618414708586890313023055}{4522366473760940656575333459867126854479891141065814030900869617085844873322532543057895876580878584512746045411554} a^{5} - \frac{521427429381100002501535521012541319480117886597751843669785389971192515062625405608380751961250736750135183762569}{6783549710641410984863000189800690281719836711598721046351304425628767309983798814586843814871317876769119068117331} a^{4} - \frac{3049971860266479112108081840388594276418513691668707578936365085110373258771469427320951365969901957368115385315085}{9044732947521881313150666919734253708959782282131628061801739234171689746645065086115791753161757169025492090823108} a^{3} + \frac{229490108180677154929056483787787669960842137927592728410200216050485108608199879421832650834469278994591953783661}{2261183236880470328287666729933563427239945570532907015450434808542922436661266271528947938290439292256373022705777} a^{2} - \frac{1067146063682741453648395726604895838011345528035848995037121389786715039722824834659493753910328010896801127650807}{6783549710641410984863000189800690281719836711598721046351304425628767309983798814586843814871317876769119068117331} a + \frac{19671028880244389914843672157768683292108049273590284923324335041079814677178328695398586822533224549762034745982}{6783549710641410984863000189800690281719836711598721046351304425628767309983798814586843814871317876769119068117331}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{75774660}$, which has order $19398312960$ (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:  $7$
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:  \( 20726.065235 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4).C_2^4$ (as 16T471):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $(C_2^2\times C_4).C_2^4$
Character table for $(C_2^2\times C_4).C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.7168.1, \(\Q(\zeta_{16})^+\), 4.4.14336.1, 8.8.3288334336.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
193Data not computed
223Data not computed