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Properties

Label 16.0.14730630136...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{8}\cdot 11^{8}$
Root discriminant $49.89$
Ramified primes $2, 5, 11$
Class number $1080$ (GRH)
Class group $[3, 3, 120]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![444959, -359648, 519304, -363464, 300592, -167312, 99796, -43784, 19101, -6368, 2224, -560, 154, -56, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 154*x^12 - 560*x^11 + 2224*x^10 - 6368*x^9 + 19101*x^8 - 43784*x^7 + 99796*x^6 - 167312*x^5 + 300592*x^4 - 363464*x^3 + 519304*x^2 - 359648*x + 444959)
 
gp: K = bnfinit(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 154*x^12 - 560*x^11 + 2224*x^10 - 6368*x^9 + 19101*x^8 - 43784*x^7 + 99796*x^6 - 167312*x^5 + 300592*x^4 - 363464*x^3 + 519304*x^2 - 359648*x + 444959, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 28 x^{14} - 56 x^{13} + 154 x^{12} - 560 x^{11} + 2224 x^{10} - 6368 x^{9} + 19101 x^{8} - 43784 x^{7} + 99796 x^{6} - 167312 x^{5} + 300592 x^{4} - 363464 x^{3} + 519304 x^{2} - 359648 x + 444959 \)

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:  \(1473063013603449241600000000=2^{44}\cdot 5^{8}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.89$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(880=2^{4}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{880}(1,·)$, $\chi_{880}(769,·)$, $\chi_{880}(329,·)$, $\chi_{880}(461,·)$, $\chi_{880}(109,·)$, $\chi_{880}(529,·)$, $\chi_{880}(661,·)$, $\chi_{880}(89,·)$, $\chi_{880}(221,·)$, $\chi_{880}(549,·)$, $\chi_{880}(681,·)$, $\chi_{880}(749,·)$, $\chi_{880}(241,·)$, $\chi_{880}(309,·)$, $\chi_{880}(441,·)$, $\chi_{880}(21,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{604280639110594} a^{14} - \frac{7}{604280639110594} a^{13} + \frac{3818268358382}{302140319555297} a^{12} - \frac{45819220300493}{604280639110594} a^{11} - \frac{198283076643}{26273071265678} a^{10} + \frac{140671753679667}{604280639110594} a^{9} + \frac{5768849342961}{604280639110594} a^{8} - \frac{162556064533795}{604280639110594} a^{7} + \frac{244180632698583}{604280639110594} a^{6} - \frac{9430265120777}{26273071265678} a^{5} + \frac{177610058933215}{604280639110594} a^{4} + \frac{88982936276159}{604280639110594} a^{3} - \frac{9739609456983}{302140319555297} a^{2} + \frac{43300332098434}{302140319555297} a + \frac{50082656530647}{604280639110594}$, $\frac{1}{93378882881120980226} a^{15} + \frac{3359}{4059951429613955662} a^{14} - \frac{1217074076840647491}{5492875463595351778} a^{13} + \frac{9176557540319609565}{46689441440560490113} a^{12} - \frac{11017549426162544097}{93378882881120980226} a^{11} - \frac{428015162076525098}{2746437731797675889} a^{10} - \frac{8833689307942000855}{46689441440560490113} a^{9} - \frac{10074916531458407548}{46689441440560490113} a^{8} - \frac{21140876945515983691}{93378882881120980226} a^{7} + \frac{1062384912300264571}{2746437731797675889} a^{6} + \frac{12896622567640325336}{46689441440560490113} a^{5} + \frac{6469449223553102102}{46689441440560490113} a^{4} - \frac{16823044353808433424}{46689441440560490113} a^{3} + \frac{3429297789153083151}{93378882881120980226} a^{2} + \frac{1024360220128284137}{5492875463595351778} a - \frac{15206231421919154500}{46689441440560490113}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{120}$, which has order $1080$ (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:  \( 12198.951274811623 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{10}, \sqrt{-11})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{10}, \sqrt{-22})\), \(\Q(\sqrt{2}, \sqrt{-55})\), \(\Q(\sqrt{5}, \sqrt{-22})\), \(\Q(\sqrt{2}, \sqrt{-11})\), \(\Q(\sqrt{5}, \sqrt{-11})\), 4.4.51200.1, \(\Q(\zeta_{16})^+\), 4.0.247808.2, 4.0.6195200.5, 8.0.37480960000.9, 8.8.2621440000.1, 8.0.38380503040000.63, 8.0.38380503040000.23, 8.0.38380503040000.59, 8.0.38380503040000.48, 8.0.61408804864.2

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$