Properties

Label 16.0.560...913.2
Degree $16$
Signature $[0, 8]$
Discriminant $5.600\times 10^{45}$
Root discriminant $723.21$
Ramified primes $17, 89$
Class number $22519584$ (GRH)
Class group $[2, 6, 1876632]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 94*x^14 - 724*x^13 + 26159*x^12 - 167999*x^11 + 2096914*x^10 - 10821888*x^9 - 25156346*x^8 - 651196200*x^7 + 18347527091*x^6 + 172806626196*x^5 + 741026109860*x^4 + 1097105545547*x^3 - 4848874441183*x^2 - 7650125420549*x + 20600166208723)
 
gp: K = bnfinit(x^16 - 3*x^15 + 94*x^14 - 724*x^13 + 26159*x^12 - 167999*x^11 + 2096914*x^10 - 10821888*x^9 - 25156346*x^8 - 651196200*x^7 + 18347527091*x^6 + 172806626196*x^5 + 741026109860*x^4 + 1097105545547*x^3 - 4848874441183*x^2 - 7650125420549*x + 20600166208723, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20600166208723, -7650125420549, -4848874441183, 1097105545547, 741026109860, 172806626196, 18347527091, -651196200, -25156346, -10821888, 2096914, -167999, 26159, -724, 94, -3, 1]);
 

\( x^{16} - 3 x^{15} + 94 x^{14} - 724 x^{13} + 26159 x^{12} - 167999 x^{11} + 2096914 x^{10} - 10821888 x^{9} - 25156346 x^{8} - 651196200 x^{7} + 18347527091 x^{6} + 172806626196 x^{5} + 741026109860 x^{4} + 1097105545547 x^{3} - 4848874441183 x^{2} - 7650125420549 x + 20600166208723 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(5600075906386661768959437042812533816731958913\)\(\medspace = 17^{15}\cdot 89^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $723.21$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $17, 89$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{47} a^{13} + \frac{16}{47} a^{12} - \frac{15}{47} a^{11} + \frac{17}{47} a^{10} + \frac{3}{47} a^{9} + \frac{3}{47} a^{7} - \frac{21}{47} a^{6} - \frac{12}{47} a^{5} - \frac{16}{47} a^{4} - \frac{6}{47} a^{3} - \frac{17}{47} a^{2} - \frac{10}{47} a - \frac{17}{47}$, $\frac{1}{47} a^{14} + \frac{11}{47} a^{12} + \frac{22}{47} a^{11} + \frac{13}{47} a^{10} - \frac{1}{47} a^{9} + \frac{3}{47} a^{8} - \frac{22}{47} a^{7} - \frac{5}{47} a^{6} - \frac{12}{47} a^{5} + \frac{15}{47} a^{4} - \frac{15}{47} a^{3} - \frac{20}{47} a^{2} + \frac{2}{47} a - \frac{10}{47}$, $\frac{1}{10374905496993658293938783470568881792274170192546609205457944903431069451646025384555210366225891} a^{15} - \frac{83585490708544564843759700150843160576821354574656730481019904993527071765015024921925113036524}{10374905496993658293938783470568881792274170192546609205457944903431069451646025384555210366225891} a^{14} + \frac{77287487576916735244598962419326309343779698487694303268623005943077359734598624187519561137563}{10374905496993658293938783470568881792274170192546609205457944903431069451646025384555210366225891} a^{13} - \frac{1556493786511927754543406983720331959965799905828694321824831128677901864403326709652174349936174}{10374905496993658293938783470568881792274170192546609205457944903431069451646025384555210366225891} a^{12} - \frac{1427610261916222157669398580582140407023643797064625082144534163393141188351477717170009000780979}{10374905496993658293938783470568881792274170192546609205457944903431069451646025384555210366225891} a^{11} + \frac{66793425281435291647237855518787207886608735118351248549987225151124745535294681166276171859311}{154849335776024750655802738366699728242898062575322525454596192588523424651433214694853886063073} a^{10} - \frac{5161444152850507702379729277783449510837331151037903643933408990322919168046416976617491556993761}{10374905496993658293938783470568881792274170192546609205457944903431069451646025384555210366225891} a^{9} - \frac{560286435115094424463575197147777441153055919034210454952789074333235478408728655354188573667984}{10374905496993658293938783470568881792274170192546609205457944903431069451646025384555210366225891} a^{8} + \frac{4653545225456375591945415773834279285247849579486944116711447412021404279159802221660536355097953}{10374905496993658293938783470568881792274170192546609205457944903431069451646025384555210366225891} a^{7} - \frac{2851254626908175641173208164535340325424939305111418901984872507448289652706756546255412674413056}{10374905496993658293938783470568881792274170192546609205457944903431069451646025384555210366225891} a^{6} - \frac{2060375081242548260588298057141908451733883491760560577668883315194998953520659616268439866344891}{10374905496993658293938783470568881792274170192546609205457944903431069451646025384555210366225891} a^{5} + \frac{1765455140163029225953842615310663384001702932798589315668699078896334968237038211334712207578362}{10374905496993658293938783470568881792274170192546609205457944903431069451646025384555210366225891} a^{4} - \frac{201559033850645005251622724283635067740548970375043347179816184565281129749513578392170682127659}{10374905496993658293938783470568881792274170192546609205457944903431069451646025384555210366225891} a^{3} - \frac{3890574509808397110867869560467204561671705272309488028853094109293911077233005368000035218635127}{10374905496993658293938783470568881792274170192546609205457944903431069451646025384555210366225891} a^{2} - \frac{3161407425195655634461418700753929439078491843820343839424683241515297471263861714769745466890960}{10374905496993658293938783470568881792274170192546609205457944903431069451646025384555210366225891} a - \frac{95651527035505748228466808077974482891615710920399088461688191348893447763074207782541537878336}{241276872023108332417181010943462367262190004477828121057161509382117894224326171733842101540137}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{1876632}$, which has order $22519584$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1263629880.47 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 1263629880.47 \cdot 22519584}{2\sqrt{5600075906386661768959437042812533816731958913}}\approx 0.461840744811$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.38915873.1, 8.8.203930643438763634753.2

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

Sibling fields

Galois closure: Deg 32

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 ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
17Data not computed
$89$89.8.7.2$x^{8} - 801$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.2$x^{8} - 801$$8$$1$$7$$C_8$$[\ ]_{8}$