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Properties

Label 16.0.21652340025...441.24
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 61^{12}$
Root discriminant $78.70$
Ramified primes $13, 61$
Class number $36$ (GRH)
Class group $[2, 18]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7078513, -3402607, 16326193, -10187697, 34363, 2368569, -1090436, 64942, 104734, -40745, 3636, 2062, -685, 64, 25, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 25*x^14 + 64*x^13 - 685*x^12 + 2062*x^11 + 3636*x^10 - 40745*x^9 + 104734*x^8 + 64942*x^7 - 1090436*x^6 + 2368569*x^5 + 34363*x^4 - 10187697*x^3 + 16326193*x^2 - 3402607*x + 7078513)
 
gp: K = bnfinit(x^16 - 7*x^15 + 25*x^14 + 64*x^13 - 685*x^12 + 2062*x^11 + 3636*x^10 - 40745*x^9 + 104734*x^8 + 64942*x^7 - 1090436*x^6 + 2368569*x^5 + 34363*x^4 - 10187697*x^3 + 16326193*x^2 - 3402607*x + 7078513, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 25 x^{14} + 64 x^{13} - 685 x^{12} + 2062 x^{11} + 3636 x^{10} - 40745 x^{9} + 104734 x^{8} + 64942 x^{7} - 1090436 x^{6} + 2368569 x^{5} + 34363 x^{4} - 10187697 x^{3} + 16326193 x^{2} - 3402607 x + 7078513 \)

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:  \(2165234002589380425486809479441=13^{8}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.70$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{60} a^{12} - \frac{3}{20} a^{11} - \frac{1}{30} a^{10} - \frac{1}{20} a^{9} + \frac{2}{5} a^{8} + \frac{1}{3} a^{7} + \frac{1}{10} a^{6} - \frac{2}{15} a^{5} + \frac{4}{15} a^{4} - \frac{13}{30} a^{3} + \frac{3}{10} a^{2} - \frac{5}{12} a + \frac{23}{60}$, $\frac{1}{1140} a^{13} + \frac{1}{285} a^{12} - \frac{59}{1140} a^{11} + \frac{211}{1140} a^{10} - \frac{33}{76} a^{9} + \frac{113}{285} a^{8} + \frac{223}{570} a^{7} + \frac{13}{114} a^{6} - \frac{97}{285} a^{5} - \frac{179}{570} a^{4} + \frac{14}{57} a^{3} + \frac{89}{1140} a^{2} + \frac{269}{570} a + \frac{59}{1140}$, $\frac{1}{2280} a^{14} - \frac{3}{380} a^{12} - \frac{11}{380} a^{11} - \frac{313}{2280} a^{10} - \frac{1}{120} a^{9} + \frac{1}{380} a^{8} + \frac{313}{1140} a^{7} - \frac{283}{1140} a^{6} - \frac{67}{380} a^{5} + \frac{43}{285} a^{4} - \frac{233}{2280} a^{3} + \frac{17}{570} a^{2} - \frac{49}{1140} a + \frac{215}{456}$, $\frac{1}{132645265557669922572914350999560} a^{15} + \frac{18409774036117872807250561333}{132645265557669922572914350999560} a^{14} + \frac{25911188038766883661375494013}{66322632778834961286457175499780} a^{13} - \frac{12781484318299866040546592321}{1745332441548288454906767776310} a^{12} + \frac{41513830856387548567272167119}{6981329766193153819627071105240} a^{11} + \frac{35639000508963112409804614825}{6632263277883496128645717549978} a^{10} + \frac{16746719596409009338520501161}{1396265953238630763925414221048} a^{9} - \frac{1417875256456762493290923958834}{5526886064902913440538097958315} a^{8} + \frac{6869956047073469864431270050487}{16580658194708740321614293874945} a^{7} - \frac{1817499853921412773287794111431}{11053772129805826881076195916630} a^{6} + \frac{20450937880656840733440480219631}{66322632778834961286457175499780} a^{5} + \frac{102955645613096911764583309813}{8843017703844661504860956733304} a^{4} + \frac{28944313909738476831480299945131}{132645265557669922572914350999560} a^{3} + \frac{22702627397853683858586180645863}{66322632778834961286457175499780} a^{2} - \frac{33300113696127302318886755585947}{132645265557669922572914350999560} a - \frac{924053320465536283997547317451}{8843017703844661504860956733304}$

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

Class group and class number

$C_{2}\times C_{18}$, which has order $36$ (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:  \( 9538292.33358 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{61}) \), 4.0.226981.1, 4.4.48373.1, 4.0.2950753.1, 8.4.24122514952861.1, 8.4.24122514952861.5, 8.0.8706943267009.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
$61$61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$