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Properties

Label 16.0.35430872509...0304.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{52}\cdot 3^{12}\cdot 23^{6}$
Root discriminant $70.28$
Ramified primes $2, 3, 23$
Class number $512$ (GRH)
Class group $[2, 4, 4, 16]$ (GRH)
Galois group $C_4.D_4$ (as 16T30)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1977337, -3324304, 3696324, -2871392, 2031636, -951840, 415672, -83624, 11832, 16184, -5588, 1968, 240, -88, 48, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 48*x^14 - 88*x^13 + 240*x^12 + 1968*x^11 - 5588*x^10 + 16184*x^9 + 11832*x^8 - 83624*x^7 + 415672*x^6 - 951840*x^5 + 2031636*x^4 - 2871392*x^3 + 3696324*x^2 - 3324304*x + 1977337)
 
gp: K = bnfinit(x^16 - 8*x^15 + 48*x^14 - 88*x^13 + 240*x^12 + 1968*x^11 - 5588*x^10 + 16184*x^9 + 11832*x^8 - 83624*x^7 + 415672*x^6 - 951840*x^5 + 2031636*x^4 - 2871392*x^3 + 3696324*x^2 - 3324304*x + 1977337, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 48 x^{14} - 88 x^{13} + 240 x^{12} + 1968 x^{11} - 5588 x^{10} + 16184 x^{9} + 11832 x^{8} - 83624 x^{7} + 415672 x^{6} - 951840 x^{5} + 2031636 x^{4} - 2871392 x^{3} + 3696324 x^{2} - 3324304 x + 1977337 \)

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:  \(354308725098774913512640610304=2^{52}\cdot 3^{12}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.28$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{3} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{739188145} a^{14} - \frac{12671869}{739188145} a^{13} + \frac{3156258}{739188145} a^{12} + \frac{270875881}{739188145} a^{11} - \frac{81187}{739188145} a^{10} - \frac{213935252}{739188145} a^{9} - \frac{5011862}{32138615} a^{8} + \frac{185758872}{739188145} a^{7} - \frac{26539567}{739188145} a^{6} + \frac{239961243}{739188145} a^{5} - \frac{221171018}{739188145} a^{4} + \frac{40682921}{147837629} a^{3} - \frac{21588376}{739188145} a^{2} + \frac{325349236}{739188145} a - \frac{43996199}{739188145}$, $\frac{1}{775764369396858502159633338700158265895} a^{15} - \frac{520248504071538208519636203137}{775764369396858502159633338700158265895} a^{14} - \frac{60006576610316216690538304021521336986}{775764369396858502159633338700158265895} a^{13} - \frac{41800885098429434666362514340460862589}{775764369396858502159633338700158265895} a^{12} - \frac{59991070502015491707864286702383136782}{775764369396858502159633338700158265895} a^{11} - \frac{320872181593108315373255095867832957178}{775764369396858502159633338700158265895} a^{10} + \frac{374023991727874395701539169008972099491}{775764369396858502159633338700158265895} a^{9} - \frac{368733568171669207192392512103934650989}{775764369396858502159633338700158265895} a^{8} + \frac{170607554999552659200479994027506120439}{775764369396858502159633338700158265895} a^{7} + \frac{57839725455026751399199147947280383514}{775764369396858502159633338700158265895} a^{6} - \frac{224747826425204394979391416129416342231}{775764369396858502159633338700158265895} a^{5} - \frac{358417186344575934907355470073238676922}{775764369396858502159633338700158265895} a^{4} - \frac{358037385414601171596357742444688000903}{775764369396858502159633338700158265895} a^{3} - \frac{69339034867207308882066895037907919653}{155152873879371700431926667740031653179} a^{2} - \frac{138838301769853098370443514833225821746}{775764369396858502159633338700158265895} a - \frac{59313351518146970830772995755270403960}{155152873879371700431926667740031653179}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{16}$, which has order $512$ (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:  \( 431260.406848 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4$ (as 16T30):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4.D_4$
Character table for $C_4.D_4$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{3})\), 4.4.423936.1, 4.4.423936.2, 8.8.718886928384.3, 8.0.404373897216.34, 8.0.6469982355456.10

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$