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Properties

Label 16.0.39228809359...4849.8
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 17^{14}$
Root discriminant $81.68$
Ramified primes $13, 17$
Class number $64$ (GRH)
Class group $[2, 4, 8]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![202696796, -184744970, 58180841, -18237738, 3318918, -4398172, 1365365, 455748, 236931, 38096, -7823, -4352, 423, -22, 12, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 12*x^14 - 22*x^13 + 423*x^12 - 4352*x^11 - 7823*x^10 + 38096*x^9 + 236931*x^8 + 455748*x^7 + 1365365*x^6 - 4398172*x^5 + 3318918*x^4 - 18237738*x^3 + 58180841*x^2 - 184744970*x + 202696796)
 
gp: K = bnfinit(x^16 - 2*x^15 + 12*x^14 - 22*x^13 + 423*x^12 - 4352*x^11 - 7823*x^10 + 38096*x^9 + 236931*x^8 + 455748*x^7 + 1365365*x^6 - 4398172*x^5 + 3318918*x^4 - 18237738*x^3 + 58180841*x^2 - 184744970*x + 202696796, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 12 x^{14} - 22 x^{13} + 423 x^{12} - 4352 x^{11} - 7823 x^{10} + 38096 x^{9} + 236931 x^{8} + 455748 x^{7} + 1365365 x^{6} - 4398172 x^{5} + 3318918 x^{4} - 18237738 x^{3} + 58180841 x^{2} - 184744970 x + 202696796 \)

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:  \(3922880935919264967742950184849=13^{12}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.68$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{32} a^{10} + \frac{3}{32} a^{9} - \frac{3}{32} a^{8} + \frac{3}{16} a^{7} - \frac{1}{4} a^{6} + \frac{3}{32} a^{4} + \frac{13}{32} a^{3} + \frac{7}{32} a^{2} - \frac{3}{16} a + \frac{3}{8}$, $\frac{1}{32} a^{11} - \frac{1}{8} a^{9} - \frac{1}{32} a^{8} + \frac{3}{16} a^{7} - \frac{1}{4} a^{6} + \frac{3}{32} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{11}{32} a^{2} - \frac{1}{16} a - \frac{1}{8}$, $\frac{1}{1088} a^{12} + \frac{1}{136} a^{11} - \frac{5}{1088} a^{10} - \frac{3}{272} a^{9} + \frac{9}{1088} a^{8} + \frac{1}{32} a^{7} + \frac{43}{1088} a^{6} + \frac{7}{272} a^{5} + \frac{53}{1088} a^{4} + \frac{5}{68} a^{3} + \frac{375}{1088} a^{2} + \frac{169}{544} a + \frac{69}{272}$, $\frac{1}{1088} a^{13} - \frac{1}{1088} a^{11} - \frac{3}{544} a^{10} + \frac{3}{1088} a^{9} - \frac{1}{272} a^{8} - \frac{25}{1088} a^{7} - \frac{11}{272} a^{6} + \frac{33}{1088} a^{5} - \frac{87}{544} a^{4} + \frac{109}{1088} a^{3} - \frac{6}{17} a^{2} - \frac{23}{136} a + \frac{47}{136}$, $\frac{1}{17408} a^{14} + \frac{7}{17408} a^{13} + \frac{3}{8704} a^{12} - \frac{195}{17408} a^{11} - \frac{105}{8704} a^{10} + \frac{1837}{17408} a^{9} + \frac{29}{544} a^{8} + \frac{2127}{17408} a^{7} + \frac{149}{8704} a^{6} + \frac{627}{17408} a^{5} + \frac{1943}{8704} a^{4} + \frac{5699}{17408} a^{3} - \frac{7557}{17408} a^{2} + \frac{2189}{8704} a + \frac{427}{4352}$, $\frac{1}{33232993696083604311504092856040425522592642382848} a^{15} - \frac{51296876462834067460968934223645593218632337}{1954881982122564959500240756237672089564273081344} a^{14} - \frac{1302680134859033958156176766806925294989141929}{16616496848041802155752046428020212761296321191424} a^{13} + \frac{8119378537853045522364160835629285524091919117}{33232993696083604311504092856040425522592642382848} a^{12} + \frac{69505633792635350523462838193766660804257562611}{16616496848041802155752046428020212761296321191424} a^{11} + \frac{233814888339238817630388928420847795837604631933}{33232993696083604311504092856040425522592642382848} a^{10} - \frac{33672432414840086183027933022062638339852056221}{4154124212010450538938011607005053190324080297856} a^{9} + \frac{13987494567607256624190297419442020559633260605}{132402365323042248253004354008129185348974670848} a^{8} + \frac{1311493596062022404599765693786477588081453238537}{16616496848041802155752046428020212761296321191424} a^{7} + \frac{6329959351565753785307969132746793712838465121571}{33232993696083604311504092856040425522592642382848} a^{6} - \frac{69914110156135455439388098061293633512151543045}{16616496848041802155752046428020212761296321191424} a^{5} - \frac{7210669661933736715078756906009347937400957665901}{33232993696083604311504092856040425522592642382848} a^{4} - \frac{12475526164477281663095521316547075678332340784509}{33232993696083604311504092856040425522592642382848} a^{3} + \frac{7763368099814942340414482127685201049953577924721}{16616496848041802155752046428020212761296321191424} a^{2} + \frac{3967676000920733290604571970086964553955661056679}{8308248424020901077876023214010106380648160595712} a - \frac{497134663590806415793323296072643985484641101687}{1038531053002612634734502901751263297581020074464}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{8}$, which has order $64$ (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:  \( 32676067.5694 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.830297.1, 4.0.3757.1, 4.0.63869.1, 8.8.1980626399884457.1, 8.0.1980626399884457.2, 8.0.689393108209.2

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$