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Properties

Label 16.0.15911238919...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{10}\cdot 29^{8}\cdot 89^{6}$
Root discriminant $158.53$
Ramified primes $2, 5, 29, 89$
Class number $5105280$ (GRH)
Class group $[2, 2, 2, 4, 159540]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T516)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![140059320025, 0, 429837223525, 0, 92746144390, 0, 7779430815, 0, 316668951, 0, 6796991, 0, 78062, 0, 449, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 449*x^14 + 78062*x^12 + 6796991*x^10 + 316668951*x^8 + 7779430815*x^6 + 92746144390*x^4 + 429837223525*x^2 + 140059320025)
 
gp: K = bnfinit(x^16 + 449*x^14 + 78062*x^12 + 6796991*x^10 + 316668951*x^8 + 7779430815*x^6 + 92746144390*x^4 + 429837223525*x^2 + 140059320025, 1)
 

Normalized defining polynomial

\( x^{16} + 449 x^{14} + 78062 x^{12} + 6796991 x^{10} + 316668951 x^{8} + 7779430815 x^{6} + 92746144390 x^{4} + 429837223525 x^{2} + 140059320025 \)

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:  \(159112389191659073629133440000000000=2^{16}\cdot 5^{10}\cdot 29^{8}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $158.53$
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:  $2, 5, 29, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{29} a^{8} + \frac{14}{29} a^{6} - \frac{6}{29} a^{4}$, $\frac{1}{29} a^{9} + \frac{14}{29} a^{7} - \frac{6}{29} a^{5}$, $\frac{1}{87} a^{10} - \frac{1}{87} a^{8} - \frac{13}{87} a^{6} - \frac{26}{87} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{87} a^{11} - \frac{1}{87} a^{9} - \frac{13}{87} a^{7} - \frac{26}{87} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3368205} a^{12} - \frac{1384}{374245} a^{10} + \frac{52252}{3368205} a^{8} + \frac{1666}{12905} a^{6} - \frac{4241}{116145} a^{4} - \frac{1}{9} a^{2} + \frac{1}{9}$, $\frac{1}{3368205} a^{13} - \frac{1384}{374245} a^{11} + \frac{52252}{3368205} a^{9} + \frac{1666}{12905} a^{7} - \frac{4241}{116145} a^{5} - \frac{1}{9} a^{3} + \frac{1}{9} a$, $\frac{1}{872657545799647518277107973275} a^{14} - \frac{5161405068812813375831}{872657545799647518277107973275} a^{12} + \frac{143240917052437633664185597}{872657545799647518277107973275} a^{10} + \frac{72444100926534371390895944}{30091639510332673044038205975} a^{8} + \frac{2533695202190599717653251089}{30091639510332673044038205975} a^{6} - \frac{31502465704532334850934182}{207528548347121883062332455} a^{4} + \frac{180119563183704939217862}{777260480700831022705365} a^{2} + \frac{3164049894377542029862}{16081251324844779780111}$, $\frac{1}{872657545799647518277107973275} a^{15} - \frac{5161405068812813375831}{872657545799647518277107973275} a^{13} + \frac{143240917052437633664185597}{872657545799647518277107973275} a^{11} + \frac{72444100926534371390895944}{30091639510332673044038205975} a^{9} + \frac{2533695202190599717653251089}{30091639510332673044038205975} a^{7} - \frac{31502465704532334850934182}{207528548347121883062332455} a^{5} + \frac{180119563183704939217862}{777260480700831022705365} a^{3} + \frac{3164049894377542029862}{16081251324844779780111} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{159540}$, which has order $5105280$ (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:  \( 13345.9350553 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T516):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.64525.2, 4.4.725.1, 4.4.2225.1, 8.8.4163475625.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.8.3$x^{8} + 2 x^{7} + 2 x^{6} + 16$$2$$4$$8$$C_2^3: C_4$$[2, 2, 2]^{4}$
2.8.8.3$x^{8} + 2 x^{7} + 2 x^{6} + 16$$2$$4$$8$$C_2^3: C_4$$[2, 2, 2]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$89$89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.8.6.2$x^{8} + 979 x^{4} + 285156$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$