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Properties

Label 16.0.11736219212...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{8}\cdot 7^{4}\cdot 13^{4}\cdot 101^{2}$
Root discriminant $49.19$
Ramified primes $2, 5, 7, 13, 101$
Class number $1228$ (GRH)
Class group $[2, 614]$ (GRH)
Galois group 16T1123

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1038209, 996728, 1745324, 452756, 871247, 39612, 237130, -21780, 41399, -7616, 5292, -812, 551, -28, 36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 36*x^14 - 28*x^13 + 551*x^12 - 812*x^11 + 5292*x^10 - 7616*x^9 + 41399*x^8 - 21780*x^7 + 237130*x^6 + 39612*x^5 + 871247*x^4 + 452756*x^3 + 1745324*x^2 + 996728*x + 1038209)
 
gp: K = bnfinit(x^16 + 36*x^14 - 28*x^13 + 551*x^12 - 812*x^11 + 5292*x^10 - 7616*x^9 + 41399*x^8 - 21780*x^7 + 237130*x^6 + 39612*x^5 + 871247*x^4 + 452756*x^3 + 1745324*x^2 + 996728*x + 1038209, 1)
 

Normalized defining polynomial

\( x^{16} + 36 x^{14} - 28 x^{13} + 551 x^{12} - 812 x^{11} + 5292 x^{10} - 7616 x^{9} + 41399 x^{8} - 21780 x^{7} + 237130 x^{6} + 39612 x^{5} + 871247 x^{4} + 452756 x^{3} + 1745324 x^{2} + 996728 x + 1038209 \)

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:  \(1173621921239636377600000000=2^{32}\cdot 5^{8}\cdot 7^{4}\cdot 13^{4}\cdot 101^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.19$
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:  $2, 5, 7, 13, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{274002739208071560322401499581621060838365389} a^{15} + \frac{16771817768786141988100275088848210075592580}{274002739208071560322401499581621060838365389} a^{14} + \frac{101047082741688394804775690908556103159085685}{274002739208071560322401499581621060838365389} a^{13} + \frac{16155088964001842081223902637078005283818544}{274002739208071560322401499581621060838365389} a^{12} - \frac{72487738884744205494043350220839759717306011}{274002739208071560322401499581621060838365389} a^{11} - \frac{50241114452298277237164490398771824145764896}{274002739208071560322401499581621060838365389} a^{10} - \frac{74713016105581138721852824425211800088782802}{274002739208071560322401499581621060838365389} a^{9} - \frac{66212878328834547763494328366573828584110352}{274002739208071560322401499581621060838365389} a^{8} + \frac{12803516930527167563367984329241501119060198}{274002739208071560322401499581621060838365389} a^{7} - \frac{103090325595637645509878663768036067764601447}{274002739208071560322401499581621060838365389} a^{6} + \frac{1980196779168077564067235917547998926246836}{16117808188710091783670676445977709461080317} a^{5} + \frac{36278486617796930477133840833905011188180641}{274002739208071560322401499581621060838365389} a^{4} - \frac{37585521090160206801614974760981442416811298}{274002739208071560322401499581621060838365389} a^{3} - \frac{7945258875530941100631668318528258499044595}{16117808188710091783670676445977709461080317} a^{2} - \frac{21227427276715550042287823611651546972161802}{274002739208071560322401499581621060838365389} a + \frac{82939986261246938371622411337564999622324148}{274002739208071560322401499581621060838365389}$

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

Class group and class number

$C_{2}\times C_{614}$, which has order $1228$ (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:  \( 3710.59482488 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1123:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1123 are not computed
Character table for t16n1123 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.432640000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
7.8.4.2$x^{8} + 49 x^{4} - 1029 x^{2} + 12005$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$101$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$
101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$