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Properties

Label 16.8.46705271798...0000.3
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{14}\cdot 61^{6}\cdot 97^{4}$
Root discriminant $169.56$
Ramified primes $2, 5, 61, 97$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T852

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-378475414519, 708955723270, -317010550824, -20861377640, 31494302817, -4945685100, -174611312, 231449980, -14651525, 985700, -894128, -103570, 4207, -600, 224, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 224*x^14 - 600*x^13 + 4207*x^12 - 103570*x^11 - 894128*x^10 + 985700*x^9 - 14651525*x^8 + 231449980*x^7 - 174611312*x^6 - 4945685100*x^5 + 31494302817*x^4 - 20861377640*x^3 - 317010550824*x^2 + 708955723270*x - 378475414519)
 
gp: K = bnfinit(x^16 + 224*x^14 - 600*x^13 + 4207*x^12 - 103570*x^11 - 894128*x^10 + 985700*x^9 - 14651525*x^8 + 231449980*x^7 - 174611312*x^6 - 4945685100*x^5 + 31494302817*x^4 - 20861377640*x^3 - 317010550824*x^2 + 708955723270*x - 378475414519, 1)
 

Normalized defining polynomial

\( x^{16} + 224 x^{14} - 600 x^{13} + 4207 x^{12} - 103570 x^{11} - 894128 x^{10} + 985700 x^{9} - 14651525 x^{8} + 231449980 x^{7} - 174611312 x^{6} - 4945685100 x^{5} + 31494302817 x^{4} - 20861377640 x^{3} - 317010550824 x^{2} + 708955723270 x - 378475414519 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(467052717980688838758400000000000000=2^{24}\cdot 5^{14}\cdot 61^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $169.56$
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:  $2, 5, 61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{13310953848668968397371318816648732693168654800988985049375919552621922362818937093161960689} a^{15} + \frac{3735256407856042114860902104216121358875117532562377376584203785578082581609035790480283844}{13310953848668968397371318816648732693168654800988985049375919552621922362818937093161960689} a^{14} - \frac{2824752950251263921934222615420085353911705883150767125608335541048522536054927144701876059}{13310953848668968397371318816648732693168654800988985049375919552621922362818937093161960689} a^{13} + \frac{109411944243734053618905373508109069598219909421504328269505700206140761661013096143919367}{13310953848668968397371318816648732693168654800988985049375919552621922362818937093161960689} a^{12} + \frac{867633362965990532249917578172935416825023566917191179267001530227473828711178687985532303}{13310953848668968397371318816648732693168654800988985049375919552621922362818937093161960689} a^{11} + \frac{5344866393080177663616034826335329534624270241617007780886603867559344941199727268143104633}{13310953848668968397371318816648732693168654800988985049375919552621922362818937093161960689} a^{10} + \frac{1158897220144140438066168108138047895805252036871858249490176652645260826440502056957564925}{13310953848668968397371318816648732693168654800988985049375919552621922362818937093161960689} a^{9} - \frac{5244812831796227572528778865184431963524217270170072181895166658141825170118249344472403504}{13310953848668968397371318816648732693168654800988985049375919552621922362818937093161960689} a^{8} - \frac{3347567575565424172547455754460077660403726561375296047879102733010173837206397077057309314}{13310953848668968397371318816648732693168654800988985049375919552621922362818937093161960689} a^{7} - \frac{6370823562016117821663197446665186234628353472120662666122252380071851620874279466368250}{13310953848668968397371318816648732693168654800988985049375919552621922362818937093161960689} a^{6} + \frac{6281062131823439634114568153905635731497351326516039700370435601049438176265081657526667055}{13310953848668968397371318816648732693168654800988985049375919552621922362818937093161960689} a^{5} + \frac{745336190201861349309618834012558397235776782210367610374154417270765421370819696372768186}{13310953848668968397371318816648732693168654800988985049375919552621922362818937093161960689} a^{4} - \frac{1756791609451885623103639423567428980877949367073319041381949869904078866211383862459044641}{13310953848668968397371318816648732693168654800988985049375919552621922362818937093161960689} a^{3} - \frac{5302682415397562680000301525261797422298128653804208933726803185674386979775499454219515583}{13310953848668968397371318816648732693168654800988985049375919552621922362818937093161960689} a^{2} + \frac{2387511969502820106015985185819613979727641541145998675935676514706659197731017890127402578}{13310953848668968397371318816648732693168654800988985049375919552621922362818937093161960689} a - \frac{2169623479547858447357784491902959260438281678889540365216214324402276764567953315556269358}{13310953848668968397371318816648732693168654800988985049375919552621922362818937093161960689}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:  \( 181820471720 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T852:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n852
Character table for t16n852 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.14884000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.4$x^{4} + 488$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
97.8.4.2$x^{8} - 912673 x^{2} + 2036173463$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$