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Properties

Label 16.0.24651048188...8976.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{68}\cdot 17^{4}$
Root discriminant $38.64$
Ramified primes $2, 17$
Class number $20$ (GRH)
Class group $[2, 10]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12028, 13184, 37920, 57952, 23040, -11360, -2448, 10016, 4004, -2496, -1472, 304, 264, -16, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 24*x^14 - 16*x^13 + 264*x^12 + 304*x^11 - 1472*x^10 - 2496*x^9 + 4004*x^8 + 10016*x^7 - 2448*x^6 - 11360*x^5 + 23040*x^4 + 57952*x^3 + 37920*x^2 + 13184*x + 12028)
 
gp: K = bnfinit(x^16 - 24*x^14 - 16*x^13 + 264*x^12 + 304*x^11 - 1472*x^10 - 2496*x^9 + 4004*x^8 + 10016*x^7 - 2448*x^6 - 11360*x^5 + 23040*x^4 + 57952*x^3 + 37920*x^2 + 13184*x + 12028, 1)
 

Normalized defining polynomial

\( x^{16} - 24 x^{14} - 16 x^{13} + 264 x^{12} + 304 x^{11} - 1472 x^{10} - 2496 x^{9} + 4004 x^{8} + 10016 x^{7} - 2448 x^{6} - 11360 x^{5} + 23040 x^{4} + 57952 x^{3} + 37920 x^{2} + 13184 x + 12028 \)

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:  \(24651048188484727368318976=2^{68}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.64$
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:  $2, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2969468} a^{14} + \frac{318875}{2969468} a^{13} - \frac{11961}{1484734} a^{12} - \frac{279719}{2969468} a^{11} + \frac{112399}{2969468} a^{10} + \frac{10967}{1484734} a^{9} + \frac{163336}{742367} a^{8} + \frac{24297}{1484734} a^{7} + \frac{189402}{742367} a^{6} - \frac{313735}{1484734} a^{5} + \frac{181135}{742367} a^{4} + \frac{213867}{1484734} a^{3} - \frac{719989}{1484734} a^{2} + \frac{349992}{742367} a - \frac{199445}{742367}$, $\frac{1}{536939691627555422821436} a^{15} + \frac{49401318885032835}{536939691627555422821436} a^{14} + \frac{8940826693401212476033}{134234922906888855705359} a^{13} - \frac{11535208591923446571736}{134234922906888855705359} a^{12} + \frac{49926041141226731762859}{536939691627555422821436} a^{11} + \frac{25113715058487823051173}{268469845813777711410718} a^{10} - \frac{1962203701029187809339}{536939691627555422821436} a^{9} + \frac{34715411628077719121671}{268469845813777711410718} a^{8} + \frac{43896587012923582616831}{268469845813777711410718} a^{7} + \frac{11579347488041761892895}{268469845813777711410718} a^{6} + \frac{95080606343561370104583}{268469845813777711410718} a^{5} + \frac{29072772625188567126854}{134234922906888855705359} a^{4} + \frac{586233796226122565167}{1203900653873442652066} a^{3} - \frac{48917305028410326659282}{134234922906888855705359} a^{2} - \frac{20963193853472051976013}{268469845813777711410718} a - \frac{6620113267172499711021}{134234922906888855705359}$

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

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (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:  \( 81246.4605619 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.4.36507222016.4, 8.0.620622774272.25, 8.4.4563402752.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$