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Properties

Label 16.8.56000759063...8913.2
Degree $16$
Signature $[8, 4]$
Discriminant $17^{15}\cdot 89^{14}$
Root discriminant $723.21$
Ramified primes $17, 89$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3596679729757, 7013223282254, 6158845500328, 2925833801882, 670089369972, 9956127740, -28256937548, -5406999958, -345111236, -20006553, -5633748, -687226, -65670, -9385, -662, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 662*x^14 - 9385*x^13 - 65670*x^12 - 687226*x^11 - 5633748*x^10 - 20006553*x^9 - 345111236*x^8 - 5406999958*x^7 - 28256937548*x^6 + 9956127740*x^5 + 670089369972*x^4 + 2925833801882*x^3 + 6158845500328*x^2 + 7013223282254*x + 3596679729757)
 
gp: K = bnfinit(x^16 - x^15 - 662*x^14 - 9385*x^13 - 65670*x^12 - 687226*x^11 - 5633748*x^10 - 20006553*x^9 - 345111236*x^8 - 5406999958*x^7 - 28256937548*x^6 + 9956127740*x^5 + 670089369972*x^4 + 2925833801882*x^3 + 6158845500328*x^2 + 7013223282254*x + 3596679729757, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 662 x^{14} - 9385 x^{13} - 65670 x^{12} - 687226 x^{11} - 5633748 x^{10} - 20006553 x^{9} - 345111236 x^{8} - 5406999958 x^{7} - 28256937548 x^{6} + 9956127740 x^{5} + 670089369972 x^{4} + 2925833801882 x^{3} + 6158845500328 x^{2} + 7013223282254 x + 3596679729757 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5600075906386661768959437042812533816731958913=17^{15}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $723.21$
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:  $17, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{47} a^{14} - \frac{21}{47} a^{13} - \frac{1}{47} a^{12} + \frac{3}{47} a^{11} + \frac{17}{47} a^{10} + \frac{15}{47} a^{9} - \frac{9}{47} a^{8} + \frac{19}{47} a^{7} - \frac{23}{47} a^{6} - \frac{4}{47} a^{5} - \frac{20}{47} a^{4} + \frac{5}{47} a^{3} + \frac{1}{47} a^{2} - \frac{1}{47} a - \frac{1}{47}$, $\frac{1}{3423403043082913273484580740260758190895687048315958820633191898101162704185985883269719} a^{15} + \frac{11428890178536827298268991822713088794585487983823816204889171599791633206860959462934}{3423403043082913273484580740260758190895687048315958820633191898101162704185985883269719} a^{14} + \frac{534532730264195412099041577124899054379867070270312058568540628281452460151557440835737}{3423403043082913273484580740260758190895687048315958820633191898101162704185985883269719} a^{13} + \frac{144702564508728189213220981056222451571202652282024350562945902219467021788416067577601}{3423403043082913273484580740260758190895687048315958820633191898101162704185985883269719} a^{12} + \frac{792866178650100466298397820038168096046151190862887386347549423356419073252533092500733}{3423403043082913273484580740260758190895687048315958820633191898101162704185985883269719} a^{11} + \frac{1254694211053871260609813726364174167498632125193814676423339491482660918467493188326402}{3423403043082913273484580740260758190895687048315958820633191898101162704185985883269719} a^{10} + \frac{1036705112518994779045157580555515492920086885873281118078479792903165767838590140776402}{3423403043082913273484580740260758190895687048315958820633191898101162704185985883269719} a^{9} - \frac{236998507593869650236187783918556072259307858621654405365093526498488484830106357804768}{3423403043082913273484580740260758190895687048315958820633191898101162704185985883269719} a^{8} - \frac{1233785452814821671622813234778560549507908915533710198227679242162773481201998059951519}{3423403043082913273484580740260758190895687048315958820633191898101162704185985883269719} a^{7} - \frac{1262717319368156572260472410989008558300723310676348018757683405605013148233005153303713}{3423403043082913273484580740260758190895687048315958820633191898101162704185985883269719} a^{6} + \frac{1336276839013285434304417525695611011090618580660544153899478378661057708387559947814080}{3423403043082913273484580740260758190895687048315958820633191898101162704185985883269719} a^{5} + \frac{688221875429479684264607075858303019533718934359005977535420796603254294302092463765485}{3423403043082913273484580740260758190895687048315958820633191898101162704185985883269719} a^{4} + \frac{437133510489986517830323349126515738489750136629676690329457358747215634742998792947396}{3423403043082913273484580740260758190895687048315958820633191898101162704185985883269719} a^{3} - \frac{837454871901801483062214011033228392170640604067720350178184219161457218821958560123193}{3423403043082913273484580740260758190895687048315958820633191898101162704185985883269719} a^{2} + \frac{32808467419814908597957382735568981282530573584212865698043288480654069025678755382904}{72838362618785388797544271069377833848844405283318272779429614853216227748637997516377} a + \frac{1000040381298555507004889669864526032509156451977888287949648437611270779278706425367544}{3423403043082913273484580740260758190895687048315958820633191898101162704185985883269719}$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  $11$
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:  \( 4692875402150000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.38915873.1, 8.8.203930643438763634753.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
17Data not computed
$89$89.8.7.4$x^{8} - 64881$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.2$x^{8} - 801$$8$$1$$7$$C_8$$[\ ]_{8}$