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Properties

Label 16.0.33662328752...4641.5
Degree $16$
Signature $[0, 8]$
Discriminant $31^{6}\cdot 41^{14}$
Root discriminant $93.42$
Ramified primes $31, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T258)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![835063, -4035828, 7076656, -6942578, 5724119, -2987980, 1479504, -432418, 129638, -12944, 1282, 562, 58, -98, 34, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 34*x^14 - 98*x^13 + 58*x^12 + 562*x^11 + 1282*x^10 - 12944*x^9 + 129638*x^8 - 432418*x^7 + 1479504*x^6 - 2987980*x^5 + 5724119*x^4 - 6942578*x^3 + 7076656*x^2 - 4035828*x + 835063)
 
gp: K = bnfinit(x^16 - 8*x^15 + 34*x^14 - 98*x^13 + 58*x^12 + 562*x^11 + 1282*x^10 - 12944*x^9 + 129638*x^8 - 432418*x^7 + 1479504*x^6 - 2987980*x^5 + 5724119*x^4 - 6942578*x^3 + 7076656*x^2 - 4035828*x + 835063, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 34 x^{14} - 98 x^{13} + 58 x^{12} + 562 x^{11} + 1282 x^{10} - 12944 x^{9} + 129638 x^{8} - 432418 x^{7} + 1479504 x^{6} - 2987980 x^{5} + 5724119 x^{4} - 6942578 x^{3} + 7076656 x^{2} - 4035828 x + 835063 \)

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:  \(33662328752972862921089838254641=31^{6}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.42$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{32} a^{12} + \frac{1}{16} a^{11} - \frac{1}{8} a^{10} + \frac{3}{32} a^{9} - \frac{3}{32} a^{8} - \frac{1}{4} a^{7} + \frac{7}{32} a^{6} + \frac{1}{8} a^{5} - \frac{1}{32} a^{4} - \frac{5}{32} a^{3} - \frac{7}{32} a^{2} - \frac{5}{32} a + \frac{11}{32}$, $\frac{1}{32} a^{13} + \frac{3}{32} a^{10} - \frac{1}{32} a^{9} - \frac{1}{16} a^{8} + \frac{7}{32} a^{7} + \frac{3}{16} a^{6} - \frac{1}{32} a^{5} + \frac{5}{32} a^{4} + \frac{3}{32} a^{3} + \frac{1}{32} a^{2} - \frac{3}{32} a + \frac{5}{16}$, $\frac{1}{8655330090652064} a^{14} - \frac{7}{8655330090652064} a^{13} + \frac{78245188049573}{8655330090652064} a^{12} - \frac{469471128297347}{8655330090652064} a^{11} - \frac{392601406297}{4327665045326032} a^{10} - \frac{5063422134387}{2163832522663016} a^{9} - \frac{304196678017713}{4327665045326032} a^{8} + \frac{1718578189422493}{8655330090652064} a^{7} - \frac{139191590299759}{1081916261331508} a^{6} + \frac{34574747188453}{270479065332877} a^{5} + \frac{1419329134586211}{8655330090652064} a^{4} + \frac{1272276841198547}{8655330090652064} a^{3} - \frac{834465081402141}{8655330090652064} a^{2} + \frac{889872480760923}{4327665045326032} a + \frac{2786791468695753}{8655330090652064}$, $\frac{1}{123277866481157347552} a^{15} + \frac{3557}{61638933240578673776} a^{14} + \frac{350039033134742701}{61638933240578673776} a^{13} - \frac{140850996520777797}{123277866481157347552} a^{12} + \frac{2270417832110308677}{123277866481157347552} a^{11} - \frac{43918659389107015}{61638933240578673776} a^{10} + \frac{6823517250738995417}{123277866481157347552} a^{9} - \frac{11242104749589963}{3852433327536167111} a^{8} + \frac{13538427016536978705}{123277866481157347552} a^{7} + \frac{26139830549715882319}{123277866481157347552} a^{6} + \frac{21880032542091461747}{123277866481157347552} a^{5} + \frac{30634159875145643977}{123277866481157347552} a^{4} - \frac{11042143777249968963}{123277866481157347552} a^{3} + \frac{22276428147851843511}{61638933240578673776} a^{2} + \frac{16350634132192654907}{61638933240578673776} a + \frac{2309341135855754951}{30819466620289336888}$

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:  $7$
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:  \( 322205509.249 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T258):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.0.194754273881.1, 8.6.5801924573188871.1, 8.2.141510355443631.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$