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Properties

Label 16.0.74787115857...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{8}\cdot 13^{6}\cdot 31^{4}$
Root discriminant $55.22$
Ramified primes $2, 5, 13, 31$
Class number $3280$ (GRH)
Class group $[2, 2, 2, 410]$ (GRH)
Galois group $(C_2\times C_4).C_2^4$ (as 16T205)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29319889, -16351756, 17099594, -16426168, 5723512, -4225052, 1914202, -431008, 363790, -19988, 37270, -424, 2168, -4, 70, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 70*x^14 - 4*x^13 + 2168*x^12 - 424*x^11 + 37270*x^10 - 19988*x^9 + 363790*x^8 - 431008*x^7 + 1914202*x^6 - 4225052*x^5 + 5723512*x^4 - 16426168*x^3 + 17099594*x^2 - 16351756*x + 29319889)
 
gp: K = bnfinit(x^16 + 70*x^14 - 4*x^13 + 2168*x^12 - 424*x^11 + 37270*x^10 - 19988*x^9 + 363790*x^8 - 431008*x^7 + 1914202*x^6 - 4225052*x^5 + 5723512*x^4 - 16426168*x^3 + 17099594*x^2 - 16351756*x + 29319889, 1)
 

Normalized defining polynomial

\( x^{16} + 70 x^{14} - 4 x^{13} + 2168 x^{12} - 424 x^{11} + 37270 x^{10} - 19988 x^{9} + 363790 x^{8} - 431008 x^{7} + 1914202 x^{6} - 4225052 x^{5} + 5723512 x^{4} - 16426168 x^{3} + 17099594 x^{2} - 16351756 x + 29319889 \)

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:  \(7478711585794844262400000000=2^{32}\cdot 5^{8}\cdot 13^{6}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.22$
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:  $2, 5, 13, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} + \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{10} - \frac{1}{8} a^{6} + \frac{1}{8} a^{2}$, $\frac{1}{35710475167682519671303728064985653712946808} a^{15} - \frac{987692333420228283224389573491994390625889}{17855237583841259835651864032492826856473404} a^{14} + \frac{645472778817202346191767196699655886492937}{35710475167682519671303728064985653712946808} a^{13} + \frac{1593542589590867603734849635919828077764935}{35710475167682519671303728064985653712946808} a^{12} - \frac{4403580156845525259040863005452579168841997}{35710475167682519671303728064985653712946808} a^{11} + \frac{466705276669917383847581264177117146381949}{4463809395960314958912966008123206714118351} a^{10} - \frac{4267747828737377987199725362667376883902821}{35710475167682519671303728064985653712946808} a^{9} + \frac{2366114710264432132549730144626890179436951}{35710475167682519671303728064985653712946808} a^{8} + \frac{310512887947512741699221665207740823306747}{35710475167682519671303728064985653712946808} a^{7} - \frac{3546274278243854382221754246762682885766461}{17855237583841259835651864032492826856473404} a^{6} + \frac{6213588584859919748924834522576827555384507}{35710475167682519671303728064985653712946808} a^{5} - \frac{4026799083725143294432674036354469141966659}{35710475167682519671303728064985653712946808} a^{4} + \frac{1218504288218535368654380143476270649681249}{35710475167682519671303728064985653712946808} a^{3} - \frac{1741162058644563657879242729925628663423035}{8927618791920629917825932016246413428236702} a^{2} + \frac{12147059524899430313908309930221447596096809}{35710475167682519671303728064985653712946808} a - \frac{2516475598488003542065047916614685112889819}{35710475167682519671303728064985653712946808}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{410}$, which has order $3280$ (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:  \( 3710.59482488 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).C_2^4$ (as 16T205):

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 $(C_2\times C_4).C_2^4$
Character table for $(C_2\times C_4).C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.432640000.1

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

Sibling fields

Degree 16 sibling: 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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
5Data not computed
$13$13.8.6.3$x^{8} + 65 x^{4} + 1352$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
31Data not computed