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Properties

Label 16.0.10530570899...6721.1
Degree $16$
Signature $[0, 8]$
Discriminant $11^{8}\cdot 53^{12}$
Root discriminant $65.15$
Ramified primes $11, 53$
Class number $18$ (GRH)
Class group $[3, 6]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![176983, -1274255, 3633675, -5547846, 5411845, -3671311, 1771924, -592576, 135908, -24160, 3372, 290, -239, 104, -26, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 26*x^14 + 104*x^13 - 239*x^12 + 290*x^11 + 3372*x^10 - 24160*x^9 + 135908*x^8 - 592576*x^7 + 1771924*x^6 - 3671311*x^5 + 5411845*x^4 - 5547846*x^3 + 3633675*x^2 - 1274255*x + 176983)
 
gp: K = bnfinit(x^16 - 2*x^15 - 26*x^14 + 104*x^13 - 239*x^12 + 290*x^11 + 3372*x^10 - 24160*x^9 + 135908*x^8 - 592576*x^7 + 1771924*x^6 - 3671311*x^5 + 5411845*x^4 - 5547846*x^3 + 3633675*x^2 - 1274255*x + 176983, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 26 x^{14} + 104 x^{13} - 239 x^{12} + 290 x^{11} + 3372 x^{10} - 24160 x^{9} + 135908 x^{8} - 592576 x^{7} + 1771924 x^{6} - 3671311 x^{5} + 5411845 x^{4} - 5547846 x^{3} + 3633675 x^{2} - 1274255 x + 176983 \)

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:  \(105305708997757955232277716721=11^{8}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.15$
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:  $11, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} + \frac{1}{11} a^{9} - \frac{2}{11} a^{5} - \frac{3}{11} a^{4} + \frac{2}{11} a^{3} - \frac{4}{11} a^{2} - \frac{4}{11} a + \frac{3}{11}$, $\frac{1}{11} a^{11} - \frac{1}{11} a^{9} - \frac{2}{11} a^{6} - \frac{1}{11} a^{5} + \frac{5}{11} a^{4} + \frac{5}{11} a^{3} - \frac{4}{11} a - \frac{3}{11}$, $\frac{1}{33} a^{12} + \frac{4}{11} a^{9} + \frac{1}{3} a^{8} + \frac{3}{11} a^{7} + \frac{10}{33} a^{6} + \frac{1}{11} a^{5} - \frac{3}{11} a^{4} - \frac{3}{11} a^{3} + \frac{14}{33} a^{2} + \frac{5}{11} a + \frac{14}{33}$, $\frac{1}{33} a^{13} - \frac{1}{33} a^{9} + \frac{3}{11} a^{8} + \frac{10}{33} a^{7} + \frac{1}{11} a^{6} + \frac{5}{11} a^{5} - \frac{2}{11} a^{4} - \frac{10}{33} a^{3} - \frac{1}{11} a^{2} - \frac{4}{33} a - \frac{1}{11}$, $\frac{1}{2533377} a^{14} + \frac{9813}{844459} a^{13} + \frac{29240}{2533377} a^{12} - \frac{5977}{844459} a^{11} - \frac{2836}{2533377} a^{10} - \frac{226574}{844459} a^{9} - \frac{229555}{2533377} a^{8} - \frac{144105}{844459} a^{7} - \frac{92791}{361911} a^{6} - \frac{210083}{844459} a^{5} + \frac{112844}{2533377} a^{4} - \frac{234643}{844459} a^{3} - \frac{410805}{844459} a^{2} - \frac{14472}{120637} a - \frac{758705}{2533377}$, $\frac{1}{4364897891331156984341715700868577} a^{15} + \frac{460083155604183389821230848}{4364897891331156984341715700868577} a^{14} - \frac{1351050630220292886312395855098}{132269633070641120737627748511169} a^{13} + \frac{5683743142248128400667882547096}{1454965963777052328113905233622859} a^{12} - \frac{43724111031950850495014854447021}{4364897891331156984341715700868577} a^{11} + \frac{134671659190563406307522127086026}{4364897891331156984341715700868577} a^{10} - \frac{1764980520303694061906777222567786}{4364897891331156984341715700868577} a^{9} + \frac{285458724773858610167294807962484}{4364897891331156984341715700868577} a^{8} + \frac{5209336202377654333541925513466}{1454965963777052328113905233622859} a^{7} + \frac{58868348343570522649467946185900}{1454965963777052328113905233622859} a^{6} + \frac{767968842090570445227826053092693}{4364897891331156984341715700868577} a^{5} - \frac{1761388059933621624663408207867260}{4364897891331156984341715700868577} a^{4} + \frac{312206700995065704020560523483510}{4364897891331156984341715700868577} a^{3} - \frac{911344880835395023466951145438680}{4364897891331156984341715700868577} a^{2} - \frac{645454763963073156986000129644238}{1454965963777052328113905233622859} a + \frac{180011234759862586141638706518772}{1454965963777052328113905233622859}$

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

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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:  \( 5314852.37338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-583}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{53})\), 4.2.30899.1 x2, 4.0.6413.1 x2, 8.0.115524532321.1, 8.2.29500764662699.1 x4

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
$53$53.4.3.1$x^{4} - 53$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.1$x^{4} - 53$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.1$x^{4} - 53$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.1$x^{4} - 53$$4$$1$$3$$C_4$$[\ ]_{4}$