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Properties

Label 16.16.2154295895...1664.3
Degree $16$
Signature $[16, 0]$
Discriminant $2^{48}\cdot 3^{8}\cdot 73^{6}\cdot 937^{4}$
Root discriminant $383.12$
Ramified primes $2, 3, 73, 937$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1191

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4107750751677169, 0, -1858790094675688, 0, 133470657752078, 0, -3668323194472, 0, 48091401945, 0, -323759520, 0, 1120320, 0, -1816, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 1816*x^14 + 1120320*x^12 - 323759520*x^10 + 48091401945*x^8 - 3668323194472*x^6 + 133470657752078*x^4 - 1858790094675688*x^2 + 4107750751677169)
 
gp: K = bnfinit(x^16 - 1816*x^14 + 1120320*x^12 - 323759520*x^10 + 48091401945*x^8 - 3668323194472*x^6 + 133470657752078*x^4 - 1858790094675688*x^2 + 4107750751677169, 1)
 

Normalized defining polynomial

\( x^{16} - 1816 x^{14} + 1120320 x^{12} - 323759520 x^{10} + 48091401945 x^{8} - 3668323194472 x^{6} + 133470657752078 x^{4} - 1858790094675688 x^{2} + 4107750751677169 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(215429589500286246610837301474983006961664=2^{48}\cdot 3^{8}\cdot 73^{6}\cdot 937^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $383.12$
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:  $2, 3, 73, 937$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{937} a^{10} + \frac{58}{937} a^{8} - \frac{332}{937} a^{6} + \frac{216}{937} a^{4} - \frac{308}{937} a^{2}$, $\frac{1}{937} a^{11} + \frac{58}{937} a^{9} - \frac{332}{937} a^{7} + \frac{216}{937} a^{5} - \frac{308}{937} a^{3}$, $\frac{1}{6473265437} a^{12} + \frac{887397}{6473265437} a^{10} + \frac{1206345940}{6473265437} a^{8} + \frac{1431957348}{6473265437} a^{6} + \frac{855586573}{6473265437} a^{4} - \frac{11040}{94637} a^{2} - \frac{7}{101}$, $\frac{1}{6473265437} a^{13} + \frac{887397}{6473265437} a^{11} + \frac{1206345940}{6473265437} a^{9} + \frac{1431957348}{6473265437} a^{7} + \frac{855586573}{6473265437} a^{5} - \frac{11040}{94637} a^{3} - \frac{7}{101} a$, $\frac{1}{887814035960311496299294980793829433231814921} a^{14} - \frac{31016194598471801007270385431521306}{887814035960311496299294980793829433231814921} a^{12} - \frac{73932455032671388757286985453079606243268}{887814035960311496299294980793829433231814921} a^{10} - \frac{202787841910246594142593109209962158954444210}{887814035960311496299294980793829433231814921} a^{8} - \frac{698991602005164744543014234844381202938065}{12161836109045362963004040832792184016874177} a^{6} + \frac{13239953572741630283953167530327816142224}{947506975411218245783665934678579971432033} a^{4} - \frac{6048104727732104094212528277882865722}{13852238642873908945536847921500854833} a^{2} - \frac{7016878411881930381826024314623514}{14783605808830212321810936949307209}$, $\frac{1}{887814035960311496299294980793829433231814921} a^{15} - \frac{31016194598471801007270385431521306}{887814035960311496299294980793829433231814921} a^{13} - \frac{73932455032671388757286985453079606243268}{887814035960311496299294980793829433231814921} a^{11} - \frac{202787841910246594142593109209962158954444210}{887814035960311496299294980793829433231814921} a^{9} - \frac{698991602005164744543014234844381202938065}{12161836109045362963004040832792184016874177} a^{7} + \frac{13239953572741630283953167530327816142224}{947506975411218245783665934678579971432033} a^{5} - \frac{6048104727732104094212528277882865722}{13852238642873908945536847921500854833} a^{3} - \frac{7016878411881930381826024314623514}{14783605808830212321810936949307209} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 2507479917520000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1191:

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 t16n1191 are not computed
Character table for t16n1191 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.10512.1, 8.8.28288548864.4

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
3Data not computed
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.8.6.1$x^{8} - 14527 x^{4} + 78021889$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
937Data not computed