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Properties

Label 16.0.11819246862...7024.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{54}\cdot 3^{8}$
Root discriminant $17.97$
Ramified primes $2, 3$
Class number $2$
Class group $[2]$
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -32, 80, 0, -224, -208, 1824, -2768, 1428, 208, -408, 128, 64, -72, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 72*x^13 + 64*x^12 + 128*x^11 - 408*x^10 + 208*x^9 + 1428*x^8 - 2768*x^7 + 1824*x^6 - 208*x^5 - 224*x^4 + 80*x^2 - 32*x + 4)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 72*x^13 + 64*x^12 + 128*x^11 - 408*x^10 + 208*x^9 + 1428*x^8 - 2768*x^7 + 1824*x^6 - 208*x^5 - 224*x^4 + 80*x^2 - 32*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 72 x^{13} + 64 x^{12} + 128 x^{11} - 408 x^{10} + 208 x^{9} + 1428 x^{8} - 2768 x^{7} + 1824 x^{6} - 208 x^{5} - 224 x^{4} + 80 x^{2} - 32 x + 4 \)

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:  \(118192468620711297024=2^{54}\cdot 3^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.97$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4}$, $\frac{1}{8} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{2} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{2} a^{7} + \frac{1}{4} a^{3}$, $\frac{1}{8} a^{12} + \frac{1}{4} a^{4}$, $\frac{1}{8} a^{13} + \frac{1}{4} a^{5}$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{13} + \frac{1}{24} a^{11} - \frac{1}{24} a^{10} + \frac{1}{24} a^{8} - \frac{1}{6} a^{7} + \frac{5}{12} a^{6} + \frac{1}{4} a^{5} - \frac{1}{6} a^{4} + \frac{5}{12} a^{3} + \frac{1}{4} a^{2} + \frac{1}{3} a + \frac{5}{12}$, $\frac{1}{415542604848} a^{15} + \frac{13775901}{2473467886} a^{14} + \frac{3199629841}{207771302424} a^{13} - \frac{2751110599}{207771302424} a^{12} - \frac{4328277691}{69257100808} a^{11} - \frac{1276788199}{25971412803} a^{10} + \frac{2957112119}{51942825606} a^{9} + \frac{926019027}{34628550404} a^{8} - \frac{13018476689}{69257100808} a^{7} - \frac{12198344264}{25971412803} a^{6} + \frac{3147142667}{103885651212} a^{5} + \frac{12499594399}{34628550404} a^{4} - \frac{6491920679}{103885651212} a^{3} - \frac{10352166718}{25971412803} a^{2} - \frac{420360841}{1236733943} a - \frac{21731552917}{51942825606}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{474429997345}{207771302424} a^{15} + \frac{478708001267}{29681614632} a^{14} - \frac{11917866494749}{207771302424} a^{13} + \frac{22106195384981}{207771302424} a^{12} - \frac{1617011758051}{51942825606} a^{11} - \frac{72796251554419}{207771302424} a^{10} + \frac{63732112724125}{103885651212} a^{9} + \frac{4959809361175}{25971412803} a^{8} - \frac{337487037648325}{103885651212} a^{7} + \frac{336371253890171}{103885651212} a^{6} - \frac{28786292600761}{103885651212} a^{5} - \frac{71657332936363}{103885651212} a^{4} - \frac{31818856669}{25971412803} a^{3} + \frac{17554248642617}{103885651212} a^{2} - \frac{242388207935}{7420403658} a - \frac{76640565745}{25971412803} \) (order $8$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 9102.34357072 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{8})\), 4.2.9216.1 x2, 4.0.4608.1 x2, 4.0.2048.1 x2, 4.2.2048.1 x2, 4.4.18432.1, 4.0.18432.2, 8.0.339738624.8, 8.0.16777216.2, 8.0.1358954496.4, 8.4.5435817984.3 x2, 8.0.5435817984.6 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$