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Properties

Label 16.12.1070491414...0000.2
Degree $16$
Signature $[12, 2]$
Discriminant $2^{32}\cdot 5^{10}\cdot 761^{5}$
Root discriminant $86.97$
Ramified primes $2, 5, 761$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1360

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1309532249, -2050188192, -514469646, 450315088, 170611937, -29825900, -9940324, 993092, -510761, -56564, 57384, 660, -773, 144, -44, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 44*x^14 + 144*x^13 - 773*x^12 + 660*x^11 + 57384*x^10 - 56564*x^9 - 510761*x^8 + 993092*x^7 - 9940324*x^6 - 29825900*x^5 + 170611937*x^4 + 450315088*x^3 - 514469646*x^2 - 2050188192*x - 1309532249)
 
gp: K = bnfinit(x^16 - 4*x^15 - 44*x^14 + 144*x^13 - 773*x^12 + 660*x^11 + 57384*x^10 - 56564*x^9 - 510761*x^8 + 993092*x^7 - 9940324*x^6 - 29825900*x^5 + 170611937*x^4 + 450315088*x^3 - 514469646*x^2 - 2050188192*x - 1309532249, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 44 x^{14} + 144 x^{13} - 773 x^{12} + 660 x^{11} + 57384 x^{10} - 56564 x^{9} - 510761 x^{8} + 993092 x^{7} - 9940324 x^{6} - 29825900 x^{5} + 170611937 x^{4} + 450315088 x^{3} - 514469646 x^{2} - 2050188192 x - 1309532249 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10704914143082750935040000000000=2^{32}\cdot 5^{10}\cdot 761^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.97$
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:  $2, 5, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{105} a^{14} - \frac{2}{21} a^{13} - \frac{1}{35} a^{12} + \frac{37}{105} a^{11} + \frac{4}{15} a^{10} - \frac{22}{105} a^{9} + \frac{32}{105} a^{8} + \frac{26}{105} a^{7} + \frac{2}{15} a^{6} + \frac{11}{35} a^{5} + \frac{3}{7} a^{4} - \frac{2}{15} a^{3} + \frac{47}{105} a^{2} + \frac{13}{35} a - \frac{1}{21}$, $\frac{1}{1821017502937407874763898551381897817879611013309900012195} a^{15} - \frac{2046970736397132953842068520501901184977302041960649266}{607005834312469291587966183793965939293203671103300004065} a^{14} - \frac{9784211281990644485063432514094647570584349292229043844}{1821017502937407874763898551381897817879611013309900012195} a^{13} + \frac{132859761979874504696039612881776068676698361191904428809}{1821017502937407874763898551381897817879611013309900012195} a^{12} + \frac{719042930326631733728023881166270859695165228209003078872}{1821017502937407874763898551381897817879611013309900012195} a^{11} - \frac{264314725070562550121523669988551030840329292860762731503}{607005834312469291587966183793965939293203671103300004065} a^{10} + \frac{227065881826156827574087048860772543143709846964290787827}{1821017502937407874763898551381897817879611013309900012195} a^{9} + \frac{696976069412653158951156479005203581454508181999271976699}{1821017502937407874763898551381897817879611013309900012195} a^{8} - \frac{248870050538642146521819262785939220968765170089890948584}{1821017502937407874763898551381897817879611013309900012195} a^{7} + \frac{44282755180049041023334096164695459380497151231874572139}{1821017502937407874763898551381897817879611013309900012195} a^{6} + \frac{14082458491473646634634530779950688257083960001501149546}{86715119187495613083995169113423705613314810157614286295} a^{5} + \frac{74676084299049851834481990619428423656702862223758574576}{1821017502937407874763898551381897817879611013309900012195} a^{4} + \frac{24262403277107819785603851240361868695158584937962822205}{121401166862493858317593236758793187858640734220660000813} a^{3} + \frac{848684018554502942413483907478839188188225295492606341101}{1821017502937407874763898551381897817879611013309900012195} a^{2} - \frac{124054536469442444825115822034677756840122746104776632117}{364203500587481574952779710276379563575922202661980002439} a + \frac{241791544879306889240866468926859992093857494530616394473}{1821017502937407874763898551381897817879611013309900012195}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 11780595119.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1360:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 65 conjugacy class representatives for t16n1360 are not computed
Character table for t16n1360 is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.1948160000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$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}$
761Data not computed