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Properties

Label 16.0.10271172687...5625.6
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{10}$
Root discriminant $27.43$
Ramified primes $5, 29$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1296, -5184, 8532, -7140, 3875, -1581, 1101, -890, 289, -245, 384, -263, 115, -50, 23, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 23*x^14 - 50*x^13 + 115*x^12 - 263*x^11 + 384*x^10 - 245*x^9 + 289*x^8 - 890*x^7 + 1101*x^6 - 1581*x^5 + 3875*x^4 - 7140*x^3 + 8532*x^2 - 5184*x + 1296)
 
gp: K = bnfinit(x^16 - 7*x^15 + 23*x^14 - 50*x^13 + 115*x^12 - 263*x^11 + 384*x^10 - 245*x^9 + 289*x^8 - 890*x^7 + 1101*x^6 - 1581*x^5 + 3875*x^4 - 7140*x^3 + 8532*x^2 - 5184*x + 1296, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 23 x^{14} - 50 x^{13} + 115 x^{12} - 263 x^{11} + 384 x^{10} - 245 x^{9} + 289 x^{8} - 890 x^{7} + 1101 x^{6} - 1581 x^{5} + 3875 x^{4} - 7140 x^{3} + 8532 x^{2} - 5184 x + 1296 \)

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:  \(102711726879931884765625=5^{12}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.43$
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:  $5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{48} a^{13} - \frac{1}{48} a^{12} + \frac{1}{24} a^{11} + \frac{1}{48} a^{10} + \frac{1}{48} a^{9} + \frac{1}{12} a^{8} - \frac{3}{16} a^{7} + \frac{1}{48} a^{6} - \frac{1}{24} a^{5} + \frac{7}{48} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{8352} a^{14} - \frac{61}{8352} a^{13} - \frac{11}{4176} a^{12} - \frac{329}{8352} a^{11} - \frac{11}{8352} a^{10} - \frac{235}{4176} a^{9} - \frac{223}{2784} a^{8} - \frac{983}{8352} a^{7} + \frac{401}{4176} a^{6} - \frac{2015}{8352} a^{5} - \frac{83}{2784} a^{4} - \frac{541}{1392} a^{3} - \frac{919}{2088} a^{2} - \frac{13}{174} a + \frac{12}{29}$, $\frac{1}{36403577904805056} a^{15} + \frac{166800029227}{18201788952402528} a^{14} - \frac{250579536593839}{36403577904805056} a^{13} - \frac{49042979001697}{1255295789820864} a^{12} + \frac{260369428657667}{18201788952402528} a^{11} + \frac{299383558289731}{36403577904805056} a^{10} + \frac{6817567731319}{237931881730752} a^{9} + \frac{2243812846554197}{18201788952402528} a^{8} + \frac{6412603553593591}{36403577904805056} a^{7} - \frac{831574453427015}{36403577904805056} a^{6} - \frac{1307832028690007}{6067262984134176} a^{5} - \frac{2107560296917577}{12134525968268352} a^{4} - \frac{3426161510087893}{9100894476201264} a^{3} - \frac{1228362009021359}{3033631492067088} a^{2} - \frac{5559138057529}{252802624338924} a + \frac{5795756044763}{84267541446308}$

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

Class group and class number

$C_{4}$, 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:  \( 119871.996246 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), 4.0.3625.1 x2, 4.0.105125.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.4.64097340625.1 x2, 8.0.11051265625.4, 8.0.11051265625.3 x2

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 8 siblings: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$