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Properties

Label 16.8.28843256040...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 181^{4}$
Root discriminant $60.08$
Ramified primes $2, 3, 5, 181$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T268)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11639401, -2933018, -6869654, 4791658, -312201, -963092, 423822, -38726, -22679, 7630, -18, 584, -511, 80, 10, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 10*x^14 + 80*x^13 - 511*x^12 + 584*x^11 - 18*x^10 + 7630*x^9 - 22679*x^8 - 38726*x^7 + 423822*x^6 - 963092*x^5 - 312201*x^4 + 4791658*x^3 - 6869654*x^2 - 2933018*x + 11639401)
 
gp: K = bnfinit(x^16 - 6*x^15 + 10*x^14 + 80*x^13 - 511*x^12 + 584*x^11 - 18*x^10 + 7630*x^9 - 22679*x^8 - 38726*x^7 + 423822*x^6 - 963092*x^5 - 312201*x^4 + 4791658*x^3 - 6869654*x^2 - 2933018*x + 11639401, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 10 x^{14} + 80 x^{13} - 511 x^{12} + 584 x^{11} - 18 x^{10} + 7630 x^{9} - 22679 x^{8} - 38726 x^{7} + 423822 x^{6} - 963092 x^{5} - 312201 x^{4} + 4791658 x^{3} - 6869654 x^{2} - 2933018 x + 11639401 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(28843256040984576000000000000=2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.08$
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:  $2, 3, 5, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{15} a^{12} - \frac{7}{15} a^{11} - \frac{1}{5} a^{10} + \frac{1}{3} a^{9} + \frac{1}{15} a^{8} - \frac{1}{3} a^{7} - \frac{1}{15} a^{6} - \frac{7}{15} a^{5} + \frac{4}{15} a^{4} + \frac{1}{5} a^{3} + \frac{4}{15} a^{2} + \frac{1}{3} a + \frac{4}{15}$, $\frac{1}{15} a^{13} - \frac{7}{15} a^{11} - \frac{1}{15} a^{10} + \frac{2}{5} a^{9} + \frac{2}{15} a^{8} - \frac{2}{5} a^{7} + \frac{1}{15} a^{6} + \frac{1}{15} a^{4} - \frac{1}{3} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{2}{15}$, $\frac{1}{2475} a^{14} - \frac{74}{2475} a^{13} + \frac{1}{275} a^{12} + \frac{1}{15} a^{11} - \frac{928}{2475} a^{10} - \frac{167}{2475} a^{9} - \frac{27}{275} a^{8} + \frac{244}{495} a^{7} - \frac{4}{15} a^{6} + \frac{11}{25} a^{5} - \frac{68}{165} a^{4} - \frac{719}{2475} a^{3} + \frac{706}{2475} a^{2} - \frac{72}{275} a - \frac{1048}{2475}$, $\frac{1}{59142290279948656007827098791183896548225} a^{15} + \frac{1938873019242964886271268635629438381}{59142290279948656007827098791183896548225} a^{14} - \frac{1150224433520335936762908545952501701261}{59142290279948656007827098791183896548225} a^{13} + \frac{40388803081403838469114991267528870708}{1314273117332192355729491084248531034405} a^{12} + \frac{16767568210073710919278927523876421655472}{59142290279948656007827098791183896548225} a^{11} - \frac{3372024403948691919968948709908076198269}{19714096759982885335942366263727965516075} a^{10} - \frac{12013875606616300256949722690459989898228}{59142290279948656007827098791183896548225} a^{9} - \frac{827175110328512895418294781690469792449}{11828458055989731201565419758236779309645} a^{8} - \frac{583913770907689555797383684263347433352}{11828458055989731201565419758236779309645} a^{7} + \frac{102246831564676197688889435395226097508}{1792190614543898666903851478520724137825} a^{6} - \frac{114980594751642812479054550638778523684}{262854623466438471145898216849706206881} a^{5} - \frac{27888230633460792726705846120254407621544}{59142290279948656007827098791183896548225} a^{4} + \frac{2225165416236957033834264104338315587686}{59142290279948656007827098791183896548225} a^{3} + \frac{16563181626331458944949561941022773565832}{59142290279948656007827098791183896548225} a^{2} + \frac{8904532090167483069335811359393440260287}{59142290279948656007827098791183896548225} a + \frac{3897473088490542271134735920419178780902}{11828458055989731201565419758236779309645}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:  \( 39923911.1096 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T268):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{15})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{60})^+\)

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

Sibling fields

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$181$181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$