↑

Properties

Label 16.0.12251765391...000.59
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 5^{8}\cdot 19^{8}$
Root discriminant $135.06$
Ramified primes $2, 3, 5, 19$
Class number $1508000$ (GRH)
Class group $[2, 2, 2, 10, 18850]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81433551601, -17556233312, 31429508788, -29370783528, 18830100752, -5211543192, 2140710428, -360080432, 108025401, -12406664, 2888720, -227640, 42490, -2128, 324, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 324*x^14 - 2128*x^13 + 42490*x^12 - 227640*x^11 + 2888720*x^10 - 12406664*x^9 + 108025401*x^8 - 360080432*x^7 + 2140710428*x^6 - 5211543192*x^5 + 18830100752*x^4 - 29370783528*x^3 + 31429508788*x^2 - 17556233312*x + 81433551601)
 
gp: K = bnfinit(x^16 - 8*x^15 + 324*x^14 - 2128*x^13 + 42490*x^12 - 227640*x^11 + 2888720*x^10 - 12406664*x^9 + 108025401*x^8 - 360080432*x^7 + 2140710428*x^6 - 5211543192*x^5 + 18830100752*x^4 - 29370783528*x^3 + 31429508788*x^2 - 17556233312*x + 81433551601, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 324 x^{14} - 2128 x^{13} + 42490 x^{12} - 227640 x^{11} + 2888720 x^{10} - 12406664 x^{9} + 108025401 x^{8} - 360080432 x^{7} + 2140710428 x^{6} - 5211543192 x^{5} + 18830100752 x^{4} - 29370783528 x^{3} + 31429508788 x^{2} - 17556233312 x + 81433551601 \)

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:  \(12251765391792386665453977600000000=2^{48}\cdot 3^{8}\cdot 5^{8}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $135.06$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4560=2^{4}\cdot 3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4560}(1,·)$, $\chi_{4560}(2051,·)$, $\chi_{4560}(1861,·)$, $\chi_{4560}(3649,·)$, $\chi_{4560}(1559,·)$, $\chi_{4560}(1369,·)$, $\chi_{4560}(3419,·)$, $\chi_{4560}(3229,·)$, $\chi_{4560}(2471,·)$, $\chi_{4560}(2281,·)$, $\chi_{4560}(4331,·)$, $\chi_{4560}(4141,·)$, $\chi_{4560}(1139,·)$, $\chi_{4560}(949,·)$, $\chi_{4560}(191,·)$, $\chi_{4560}(3839,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{19} a^{4} - \frac{2}{19} a^{3} - \frac{1}{19} a^{2} + \frac{2}{19} a + \frac{1}{19}$, $\frac{1}{19} a^{5} - \frac{5}{19} a^{3} + \frac{5}{19} a + \frac{2}{19}$, $\frac{1}{19} a^{6} + \frac{9}{19} a^{3} - \frac{7}{19} a + \frac{5}{19}$, $\frac{1}{19} a^{7} - \frac{1}{19} a^{3} + \frac{2}{19} a^{2} + \frac{6}{19} a - \frac{9}{19}$, $\frac{1}{361} a^{8} - \frac{4}{361} a^{7} + \frac{2}{361} a^{6} + \frac{8}{361} a^{5} - \frac{5}{361} a^{4} - \frac{8}{361} a^{3} + \frac{2}{361} a^{2} + \frac{4}{361} a + \frac{1}{361}$, $\frac{1}{361} a^{9} + \frac{5}{361} a^{7} - \frac{3}{361} a^{6} + \frac{8}{361} a^{5} - \frac{9}{361} a^{4} - \frac{163}{361} a^{3} + \frac{31}{361} a^{2} - \frac{154}{361} a + \frac{80}{361}$, $\frac{1}{361} a^{10} - \frac{2}{361} a^{7} - \frac{2}{361} a^{6} + \frac{8}{361} a^{5} - \frac{5}{361} a^{4} - \frac{100}{361} a^{3} + \frac{26}{361} a^{2} + \frac{136}{361} a + \frac{52}{361}$, $\frac{1}{361} a^{11} + \frac{9}{361} a^{7} - \frac{7}{361} a^{6} - \frac{8}{361} a^{5} + \frac{4}{361} a^{4} + \frac{48}{361} a^{3} + \frac{64}{361} a^{2} + \frac{79}{361} a + \frac{173}{361}$, $\frac{1}{665323} a^{12} - \frac{6}{665323} a^{11} - \frac{314}{665323} a^{10} - \frac{218}{665323} a^{9} - \frac{277}{665323} a^{8} + \frac{17094}{665323} a^{7} - \frac{9592}{665323} a^{6} - \frac{2046}{665323} a^{5} + \frac{10287}{665323} a^{4} - \frac{188490}{665323} a^{3} - \frac{248796}{665323} a^{2} - \frac{174775}{665323} a - \frac{271015}{665323}$, $\frac{1}{665323} a^{13} - \frac{350}{665323} a^{11} - \frac{259}{665323} a^{10} + \frac{258}{665323} a^{9} + \frac{688}{665323} a^{8} + \frac{17409}{665323} a^{7} + \frac{6750}{665323} a^{6} + \frac{14598}{665323} a^{5} - \frac{464}{35017} a^{4} + \frac{4357}{665323} a^{3} + \frac{53811}{665323} a^{2} - \frac{326288}{665323} a + \frac{3122}{665323}$, $\frac{1}{9077724187758515309363} a^{14} - \frac{1}{1296817741108359329909} a^{13} + \frac{5179184791310604}{9077724187758515309363} a^{12} - \frac{31075108747863533}{9077724187758515309363} a^{11} - \frac{8248247541625030228}{9077724187758515309363} a^{10} - \frac{8766007891557837607}{9077724187758515309363} a^{9} + \frac{362678727585229386}{394683660337326752581} a^{8} - \frac{56550372929528182896}{9077724187758515309363} a^{7} - \frac{26091059681678997695}{1296817741108359329909} a^{6} - \frac{148117166392355287626}{9077724187758515309363} a^{5} + \frac{19358907393865746260}{9077724187758515309363} a^{4} - \frac{4035669608530609470335}{9077724187758515309363} a^{3} + \frac{1546765574053975989635}{9077724187758515309363} a^{2} + \frac{129165242147726509435}{1296817741108359329909} a - \frac{4004051800418262192171}{9077724187758515309363}$, $\frac{1}{648556924345228783014117720803} a^{15} + \frac{35722433}{648556924345228783014117720803} a^{14} + \frac{215434282453162137233656}{648556924345228783014117720803} a^{13} - \frac{240509619287901533309870}{648556924345228783014117720803} a^{12} - \frac{167400388534328500380174733}{648556924345228783014117720803} a^{11} - \frac{601181198796293573804961934}{648556924345228783014117720803} a^{10} - \frac{337598307593781446118122217}{648556924345228783014117720803} a^{9} + \frac{60253413561322638279946880}{648556924345228783014117720803} a^{8} - \frac{6993446185169942379545863818}{648556924345228783014117720803} a^{7} + \frac{9576551772671418166713356735}{648556924345228783014117720803} a^{6} - \frac{22959899191707768379584545}{8209581320825680797647059757} a^{5} + \frac{75874781262105082204308016}{4028303877920675670895141123} a^{4} + \frac{228501733677301823793645166073}{648556924345228783014117720803} a^{3} - \frac{4445558599016602213017364132}{648556924345228783014117720803} a^{2} - \frac{7888873159340989838227741898}{648556924345228783014117720803} a - \frac{18237340498025131754045048649}{648556924345228783014117720803}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{10}\times C_{18850}$, which has order $1508000$ (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:  \( 15197.42445606848 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{30}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{6}, \sqrt{10})\), 4.0.739328.2, 4.0.166348800.2, 4.0.6653952.2, 4.0.18483200.2, 8.8.3317760000.1, 8.0.110687693045760000.10, 8.0.110687693045760000.14, 8.0.177100308873216.51, 8.0.110687693045760000.18, 8.0.341628682240000.69, 8.0.27671923261440000.256

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$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}$
$19$19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$