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Properties

Label 16.0.43482378701...6896.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{54}\cdot 17^{6}$
Root discriminant $30.02$
Ramified primes $2, 17$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37636, -108640, 151152, -132224, 77392, -28720, 5872, -2064, 3236, -2672, 1080, -160, -8, -24, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 24*x^13 - 8*x^12 - 160*x^11 + 1080*x^10 - 2672*x^9 + 3236*x^8 - 2064*x^7 + 5872*x^6 - 28720*x^5 + 77392*x^4 - 132224*x^3 + 151152*x^2 - 108640*x + 37636)
 
gp: K = bnfinit(x^16 - 8*x^15 + 24*x^14 - 24*x^13 - 8*x^12 - 160*x^11 + 1080*x^10 - 2672*x^9 + 3236*x^8 - 2064*x^7 + 5872*x^6 - 28720*x^5 + 77392*x^4 - 132224*x^3 + 151152*x^2 - 108640*x + 37636, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 24 x^{14} - 24 x^{13} - 8 x^{12} - 160 x^{11} + 1080 x^{10} - 2672 x^{9} + 3236 x^{8} - 2064 x^{7} + 5872 x^{6} - 28720 x^{5} + 77392 x^{4} - 132224 x^{3} + 151152 x^{2} - 108640 x + 37636 \)

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:  \(434823787016118543056896=2^{54}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.02$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4}$, $\frac{1}{8} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a$, $\frac{1}{8} a^{10} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} + \frac{1}{4} a^{3}$, $\frac{1}{8} a^{12} + \frac{1}{4} a^{4}$, $\frac{1}{8} a^{13} + \frac{1}{4} a^{5}$, $\frac{1}{8} a^{14} + \frac{1}{4} a^{6}$, $\frac{1}{47749943163363168905744} a^{15} + \frac{51905232701657559562}{2984371447710198056609} a^{14} - \frac{38327691433877934871}{2984371447710198056609} a^{13} + \frac{16722794585093785159}{3410710225954512064696} a^{12} - \frac{1018741289429068039573}{23874971581681584452872} a^{11} - \frac{641471005892091674461}{11937485790840792226436} a^{10} - \frac{247704554250298674523}{11937485790840792226436} a^{9} + \frac{265386446565285882707}{5968742895420396113218} a^{8} - \frac{3494582759025109190231}{23874971581681584452872} a^{7} + \frac{1314048623991654393746}{2984371447710198056609} a^{6} + \frac{273426326856169291905}{5968742895420396113218} a^{5} - \frac{259796560431452475451}{11937485790840792226436} a^{4} - \frac{251986558996851411105}{11937485790840792226436} a^{3} - \frac{1090426460897741266641}{5968742895420396113218} a^{2} - \frac{1445422165528017432015}{5968742895420396113218} a - \frac{6813534292346699044}{30766715955775237697}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 25351.9210048 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.4352.1, \(\Q(\zeta_{16})^+\), 4.4.34816.1, 8.0.20606615552.1, 8.0.82426462208.4, 8.8.4848615424.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.3.4$x^{4} + 459$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.4$x^{4} + 459$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$