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Properties

Label 16.4.69979484620...7136.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{24}\cdot 3^{14}\cdot 11^{8}\cdot 2017^{2}$
Root discriminant $63.51$
Ramified primes $2, 3, 11, 2017$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T984

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4748368, -4249952, 6378576, -3126928, 1805012, -567384, 145636, -53780, -23691, 8368, -8750, 1908, -574, -16, 18, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 18*x^14 - 16*x^13 - 574*x^12 + 1908*x^11 - 8750*x^10 + 8368*x^9 - 23691*x^8 - 53780*x^7 + 145636*x^6 - 567384*x^5 + 1805012*x^4 - 3126928*x^3 + 6378576*x^2 - 4249952*x + 4748368)
 
gp: K = bnfinit(x^16 - 8*x^15 + 18*x^14 - 16*x^13 - 574*x^12 + 1908*x^11 - 8750*x^10 + 8368*x^9 - 23691*x^8 - 53780*x^7 + 145636*x^6 - 567384*x^5 + 1805012*x^4 - 3126928*x^3 + 6378576*x^2 - 4249952*x + 4748368, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 18 x^{14} - 16 x^{13} - 574 x^{12} + 1908 x^{11} - 8750 x^{10} + 8368 x^{9} - 23691 x^{8} - 53780 x^{7} + 145636 x^{6} - 567384 x^{5} + 1805012 x^{4} - 3126928 x^{3} + 6378576 x^{2} - 4249952 x + 4748368 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(69979484620810119540570587136=2^{24}\cdot 3^{14}\cdot 11^{8}\cdot 2017^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.51$
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:  $2, 3, 11, 2017$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{24} a^{12} - \frac{1}{12} a^{11} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} - \frac{1}{12} a^{6} - \frac{1}{6} a^{5} + \frac{1}{24} a^{4} + \frac{1}{4} a^{3} + \frac{1}{12} a^{2} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{24} a^{13} - \frac{1}{12} a^{9} - \frac{1}{4} a^{7} + \frac{3}{8} a^{5} - \frac{1}{3} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{145392} a^{14} - \frac{79}{12116} a^{13} - \frac{397}{36348} a^{12} - \frac{502}{9087} a^{11} + \frac{905}{24232} a^{10} - \frac{2075}{36348} a^{9} + \frac{1519}{72696} a^{8} + \frac{137}{18174} a^{7} - \frac{19639}{145392} a^{6} - \frac{2869}{9087} a^{5} - \frac{24725}{72696} a^{4} + \frac{209}{3029} a^{3} - \frac{917}{36348} a^{2} - \frac{1663}{18174} a - \frac{844}{3029}$, $\frac{1}{28017993875891076011476670957793311013648} a^{15} - \frac{3045966742806496800497024866844959}{3113110430654564001275185661977034557072} a^{14} - \frac{9373324524313631363179581287410560113}{1556555215327282000637592830988517278536} a^{13} + \frac{16025674885386996482390778709484976931}{14008996937945538005738335478896655506824} a^{12} - \frac{184268341730766774171400185618054363613}{4669665645981846001912778492965551835608} a^{11} + \frac{185886280115068239649453243711519465003}{4669665645981846001912778492965551835608} a^{10} - \frac{575248095731377076810145483585586894045}{14008996937945538005738335478896655506824} a^{9} - \frac{36214336111447920367964488020812563949}{4669665645981846001912778492965551835608} a^{8} - \frac{508790538407941980991623082140800750617}{9339331291963692003825556985931103671216} a^{7} + \frac{430574679247711035814874351755851226597}{2155230298145467385498205458291793154896} a^{6} + \frac{351047436529823189322706885957047377443}{2334832822990923000956389246482775917804} a^{5} + \frac{90577017930924217596719202621103875586}{583708205747730750239097311620693979451} a^{4} + \frac{1997403088169359797105878427232745680955}{7004498468972769002869167739448327753412} a^{3} - \frac{115284802133475516095487386560913796521}{389138803831820500159398207747129319634} a^{2} + \frac{299387186309978990271063112941172746977}{1167416411495461500478194623241387958902} a + \frac{1548124419519703457603359033950626699237}{3502249234486384501434583869724163876706}$

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:  $9$
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:  \( 60049258.9118 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T984:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n984
Character table for t16n984 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{11}) \), 4.4.4752.1 x2, 4.4.13068.1 x2, \(\Q(\sqrt{3}, \sqrt{11})\), 8.8.2732361984.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.16.5$x^{8} + 4 x^{6} + 40 x^{2} + 4$$4$$2$$16$$D_4$$[2, 3]^{2}$
3Data not computed
$11$11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
2017Data not computed