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Properties

Label 16.0.11576669171...7137.2
Degree $16$
Signature $[0, 8]$
Discriminant $17^{13}\cdot 43^{8}$
Root discriminant $65.54$
Ramified primes $17, 43$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![775541, 475135, -373724, -309188, 426206, -340070, 131882, -7080, -6476, 1464, 3302, -1420, -188, 168, -4, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 4*x^14 + 168*x^13 - 188*x^12 - 1420*x^11 + 3302*x^10 + 1464*x^9 - 6476*x^8 - 7080*x^7 + 131882*x^6 - 340070*x^5 + 426206*x^4 - 309188*x^3 - 373724*x^2 + 475135*x + 775541)
 
gp: K = bnfinit(x^16 - 8*x^15 - 4*x^14 + 168*x^13 - 188*x^12 - 1420*x^11 + 3302*x^10 + 1464*x^9 - 6476*x^8 - 7080*x^7 + 131882*x^6 - 340070*x^5 + 426206*x^4 - 309188*x^3 - 373724*x^2 + 475135*x + 775541, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 4 x^{14} + 168 x^{13} - 188 x^{12} - 1420 x^{11} + 3302 x^{10} + 1464 x^{9} - 6476 x^{8} - 7080 x^{7} + 131882 x^{6} - 340070 x^{5} + 426206 x^{4} - 309188 x^{3} - 373724 x^{2} + 475135 x + 775541 \)

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:  \(115766691713731939356121017137=17^{13}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.54$
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:  $17, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{17} a^{8} - \frac{4}{17} a^{7} + \frac{7}{17} a^{6} - \frac{7}{17} a^{5} - \frac{2}{17} a^{4} - \frac{6}{17} a^{3} - \frac{7}{17} a^{2} + \frac{1}{17} a + \frac{1}{17}$, $\frac{1}{17} a^{9} + \frac{8}{17} a^{7} + \frac{4}{17} a^{6} + \frac{4}{17} a^{5} + \frac{3}{17} a^{4} + \frac{3}{17} a^{3} + \frac{7}{17} a^{2} + \frac{5}{17} a + \frac{4}{17}$, $\frac{1}{17} a^{10} + \frac{2}{17} a^{7} - \frac{1}{17} a^{6} + \frac{8}{17} a^{5} + \frac{2}{17} a^{4} + \frac{4}{17} a^{3} - \frac{7}{17} a^{2} - \frac{4}{17} a - \frac{8}{17}$, $\frac{1}{17} a^{11} + \frac{7}{17} a^{7} - \frac{6}{17} a^{6} - \frac{1}{17} a^{5} + \frac{8}{17} a^{4} + \frac{5}{17} a^{3} - \frac{7}{17} a^{2} + \frac{7}{17} a - \frac{2}{17}$, $\frac{1}{3757} a^{12} - \frac{6}{3757} a^{11} + \frac{8}{3757} a^{10} + \frac{15}{3757} a^{9} - \frac{70}{3757} a^{8} + \frac{124}{3757} a^{7} - \frac{537}{3757} a^{6} + \frac{1306}{3757} a^{5} + \frac{1685}{3757} a^{4} - \frac{1708}{3757} a^{3} - \frac{1607}{3757} a^{2} + \frac{789}{3757} a + \frac{13}{289}$, $\frac{1}{3757} a^{13} - \frac{28}{3757} a^{11} + \frac{63}{3757} a^{10} + \frac{20}{3757} a^{9} - \frac{75}{3757} a^{8} - \frac{677}{3757} a^{7} - \frac{369}{3757} a^{6} + \frac{460}{3757} a^{5} + \frac{446}{3757} a^{4} + \frac{1847}{3757} a^{3} + \frac{67}{289} a^{2} + \frac{1367}{3757} a + \frac{95}{289}$, $\frac{1}{24840181616825861} a^{14} - \frac{7}{24840181616825861} a^{13} - \frac{1547422863749}{24840181616825861} a^{12} + \frac{9284537182585}{24840181616825861} a^{11} - \frac{5550990080310}{1461187153930933} a^{10} + \frac{386725899319154}{24840181616825861} a^{9} + \frac{6954851463174}{1910783201294297} a^{8} - \frac{2579877762991251}{24840181616825861} a^{7} + \frac{4087890523186330}{24840181616825861} a^{6} - \frac{1711980530957900}{24840181616825861} a^{5} + \frac{5607100544209458}{24840181616825861} a^{4} - \frac{12093144437779481}{24840181616825861} a^{3} + \frac{5646668144424859}{24840181616825861} a^{2} + \frac{652834268614009}{24840181616825861} a - \frac{46009341332986}{112399011840841}$, $\frac{1}{86021548939067956643} a^{15} + \frac{1724}{86021548939067956643} a^{14} + \frac{8005229244139337}{86021548939067956643} a^{13} + \frac{229772448275204}{7820140812642541513} a^{12} + \frac{364076906702773942}{86021548939067956643} a^{11} - \frac{406001981326755823}{86021548939067956643} a^{10} - \frac{1940253264315694851}{86021548939067956643} a^{9} - \frac{1557025836941626177}{86021548939067956643} a^{8} - \frac{36100329588646955782}{86021548939067956643} a^{7} + \frac{29969972992952313570}{86021548939067956643} a^{6} + \frac{7050573518159299578}{86021548939067956643} a^{5} - \frac{2200558853428196776}{7820140812642541513} a^{4} + \frac{40771457758596258172}{86021548939067956643} a^{3} + \frac{27827083651870258806}{86021548939067956643} a^{2} - \frac{27607355530248511265}{86021548939067956643} a - \frac{1739286817061970700}{6617042226082150511}$

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

Class group and class number

$C_{4}$, 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:  $7$
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:  \( 32889782.1041 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-43}) \), 4.0.31433.1, 8.0.4854208531457.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$17$17.8.7.7$x^{8} + 51$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.6.3$x^{8} - 17 x^{4} + 867$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
43Data not computed