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Properties

Label 16.0.67658452310...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{8}\cdot 5^{8}\cdot 7^{4}$
Root discriminant $35.64$
Ramified primes $2, 3, 5, 7$
Class number $40$ (GRH)
Class group $[2, 2, 10]$ (GRH)
Galois group $C_2 \times (C_4\times C_2):C_2$ (as 16T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2521, 2412, 7220, 6292, 10494, 3708, 2952, -2324, 18, -828, 460, -156, 98, -44, 16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 16*x^14 - 44*x^13 + 98*x^12 - 156*x^11 + 460*x^10 - 828*x^9 + 18*x^8 - 2324*x^7 + 2952*x^6 + 3708*x^5 + 10494*x^4 + 6292*x^3 + 7220*x^2 + 2412*x + 2521)
 
gp: K = bnfinit(x^16 - 4*x^15 + 16*x^14 - 44*x^13 + 98*x^12 - 156*x^11 + 460*x^10 - 828*x^9 + 18*x^8 - 2324*x^7 + 2952*x^6 + 3708*x^5 + 10494*x^4 + 6292*x^3 + 7220*x^2 + 2412*x + 2521, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 16 x^{14} - 44 x^{13} + 98 x^{12} - 156 x^{11} + 460 x^{10} - 828 x^{9} + 18 x^{8} - 2324 x^{7} + 2952 x^{6} + 3708 x^{5} + 10494 x^{4} + 6292 x^{3} + 7220 x^{2} + 2412 x + 2521 \)

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:  \(6765845231016345600000000=2^{40}\cdot 3^{8}\cdot 5^{8}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.64$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{2} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{3} a^{5} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{3815572732780284670926582} a^{15} - \frac{28719299493238921893919}{1907786366390142335463291} a^{14} + \frac{49415451854767582530227}{635928788796714111821097} a^{13} - \frac{128217816814649055340193}{3815572732780284670926582} a^{12} - \frac{133195021411347756209854}{1907786366390142335463291} a^{11} + \frac{748187347188897630593}{19769806905597329901174} a^{10} - \frac{409018753779876147181880}{1907786366390142335463291} a^{9} + \frac{213515425129032376456921}{3815572732780284670926582} a^{8} + \frac{507814608937739677563005}{3815572732780284670926582} a^{7} + \frac{240158998450039227340192}{1907786366390142335463291} a^{6} - \frac{842944424670935582800975}{1907786366390142335463291} a^{5} + \frac{1343789993619805034412737}{3815572732780284670926582} a^{4} + \frac{124765687912456011752045}{1907786366390142335463291} a^{3} + \frac{109404226453407238173029}{1271857577593428223642194} a^{2} + \frac{124887754441708047131059}{1907786366390142335463291} a - \frac{1573925968464148985808809}{3815572732780284670926582}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{10}$, which has order $40$ (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:  \( 15197.4244561 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4:C_2$ (as 16T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2 \times (C_4\times C_2):C_2$
Character table for $C_2 \times (C_4\times C_2):C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{10})\), 8.8.3317760000.1, 8.0.2601123840000.4, 8.0.2601123840000.12

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

Sibling fields

Galois closure: data not computed
Degree 16 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 R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{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}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$