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Properties

Label 16.0.26268600651...000.22
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{8}\cdot 5^{8}\cdot 17^{12}$
Root discriminant $218.13$
Ramified primes $2, 3, 5, 17$
Class number $18432000$ (GRH)
Class group $[4, 4, 4, 20, 120, 120]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![157013824831, -12798160796, 48974600382, -4555750032, 6577657217, -703589400, 539585678, -58332164, 31197750, -2552476, 1076678, -61080, 21077, -768, 222, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 222*x^14 - 768*x^13 + 21077*x^12 - 61080*x^11 + 1076678*x^10 - 2552476*x^9 + 31197750*x^8 - 58332164*x^7 + 539585678*x^6 - 703589400*x^5 + 6577657217*x^4 - 4555750032*x^3 + 48974600382*x^2 - 12798160796*x + 157013824831)
 
gp: K = bnfinit(x^16 - 4*x^15 + 222*x^14 - 768*x^13 + 21077*x^12 - 61080*x^11 + 1076678*x^10 - 2552476*x^9 + 31197750*x^8 - 58332164*x^7 + 539585678*x^6 - 703589400*x^5 + 6577657217*x^4 - 4555750032*x^3 + 48974600382*x^2 - 12798160796*x + 157013824831, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 222 x^{14} - 768 x^{13} + 21077 x^{12} - 61080 x^{11} + 1076678 x^{10} - 2552476 x^{9} + 31197750 x^{8} - 58332164 x^{7} + 539585678 x^{6} - 703589400 x^{5} + 6577657217 x^{4} - 4555750032 x^{3} + 48974600382 x^{2} - 12798160796 x + 157013824831 \)

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:  \(26268600651362281870291736985600000000=2^{44}\cdot 3^{8}\cdot 5^{8}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $218.13$
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:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4080=2^{4}\cdot 3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{4080}(1,·)$, $\chi_{4080}(3841,·)$, $\chi_{4080}(1801,·)$, $\chi_{4080}(2189,·)$, $\chi_{4080}(3149,·)$, $\chi_{4080}(1441,·)$, $\chi_{4080}(149,·)$, $\chi_{4080}(3481,·)$, $\chi_{4080}(2401,·)$, $\chi_{4080}(2789,·)$, $\chi_{4080}(361,·)$, $\chi_{4080}(749,·)$, $\chi_{4080}(2549,·)$, $\chi_{4080}(2041,·)$, $\chi_{4080}(509,·)$, $\chi_{4080}(1109,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{15} a^{8} - \frac{2}{15} a^{7} + \frac{4}{15} a^{6} - \frac{1}{15} a^{5} + \frac{2}{5} a^{4} + \frac{1}{15} a^{3} + \frac{4}{15} a^{2} + \frac{2}{15} a + \frac{1}{15}$, $\frac{1}{15} a^{9} + \frac{7}{15} a^{6} + \frac{4}{15} a^{5} - \frac{2}{15} a^{4} + \frac{2}{5} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{2}{15}$, $\frac{1}{15} a^{10} + \frac{7}{15} a^{7} + \frac{4}{15} a^{6} - \frac{2}{15} a^{5} + \frac{2}{5} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{2}{15} a$, $\frac{1}{15} a^{11} + \frac{1}{5} a^{7} - \frac{2}{15} a^{5} - \frac{2}{15} a^{4} - \frac{2}{15} a^{3} + \frac{4}{15} a^{2} + \frac{1}{15} a - \frac{7}{15}$, $\frac{1}{120} a^{12} - \frac{1}{30} a^{10} - \frac{1}{30} a^{9} - \frac{1}{60} a^{8} - \frac{2}{5} a^{7} + \frac{9}{20} a^{6} + \frac{1}{3} a^{5} - \frac{2}{5} a^{4} - \frac{1}{6} a^{3} - \frac{1}{30} a^{2} + \frac{17}{120}$, $\frac{1}{120} a^{13} - \frac{1}{30} a^{11} - \frac{1}{30} a^{10} - \frac{1}{60} a^{9} - \frac{7}{20} a^{7} - \frac{1}{15} a^{6} + \frac{1}{5} a^{5} + \frac{7}{30} a^{4} + \frac{11}{30} a^{3} - \frac{2}{5} a^{2} - \frac{7}{120} a + \frac{2}{5}$, $\frac{1}{13741287954601070021520} a^{14} + \frac{5730568813671036499}{6870643977300535010760} a^{13} - \frac{211629689220154403}{916085863640071334768} a^{12} + \frac{45600598693600703581}{3435321988650267505380} a^{11} - \frac{49525026909235840881}{2290214659100178336920} a^{10} + \frac{6477338261330170029}{286276832387522292115} a^{9} + \frac{20986031212192276739}{1717660994325133752690} a^{8} + \frac{395354986846805874489}{1145107329550089168460} a^{7} - \frac{192944054507729136437}{2290214659100178336920} a^{6} - \frac{679792800205172551061}{3435321988650267505380} a^{5} + \frac{456689907027068546101}{3435321988650267505380} a^{4} - \frac{278925769219655326985}{687064397730053501076} a^{3} - \frac{1279679497725640477459}{13741287954601070021520} a^{2} - \frac{1735418588355707717533}{6870643977300535010760} a - \frac{6146495902861947565753}{13741287954601070021520}$, $\frac{1}{511932385444372283247532862392596450254836080} a^{15} + \frac{9565113724916835010607}{511932385444372283247532862392596450254836080} a^{14} - \frac{354092087224015054116511091870778116997809}{511932385444372283247532862392596450254836080} a^{13} + \frac{14399268455632117250795760071378626342823}{34128825696291485549835524159506430016989072} a^{12} - \frac{7459478552207985498730348589395345827288733}{255966192722186141623766431196298225127418040} a^{11} - \frac{7354482624217272805187617414256802105453811}{255966192722186141623766431196298225127418040} a^{10} + \frac{4218562659067051895104936077082269529963441}{127983096361093070811883215598149112563709020} a^{9} - \frac{558396282501570175344822921942051674555189}{31995774090273267702970803899537278140927255} a^{8} + \frac{3262663170485379299220280676590588624621431}{17064412848145742774917762079753215008494536} a^{7} - \frac{7205383791088102424332556590588273929211}{134365455497210573030848520313017441011768} a^{6} - \frac{940768737443947201093976096079662826084874}{31995774090273267702970803899537278140927255} a^{5} + \frac{1862570217017911872254336955577190582647167}{4266103212036435693729440519938303752123634} a^{4} - \frac{47213739462236440332656504464476426313238459}{102386477088874456649506572478519290050967216} a^{3} + \frac{16746240065878862528504034467747038694107949}{34128825696291485549835524159506430016989072} a^{2} - \frac{197451976355582776498195086028254496921057981}{511932385444372283247532862392596450254836080} a + \frac{27086293509465265473595507082428279263107747}{170644128481457427749177620797532150084945360}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}\times C_{20}\times C_{120}\times C_{120}$, which has order $18432000$ (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:  \( 103646.40189541418 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), 4.0.133171200.5, \(\Q(\sqrt{2}, \sqrt{17})\), 4.0.460800.2, 4.4.4913.1, 4.4.314432.1, 4.0.2263910400.16, 4.0.2263910400.15, 8.0.17734568509440000.484, 8.8.98867482624.1, 8.0.5125290299228160000.30

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

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$