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Properties

Label 16.0.26252865470...9681.4
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 101^{12}$
Root discriminant $218.12$
Ramified primes $13, 101$
Class number $5000$ (GRH)
Class group $[5, 5, 10, 20]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![71134267, -15162339, -17975866, -58934929, 54351103, -24590405, 19646115, -4701297, 2836720, -440625, 204969, -21126, 7718, -476, 141, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 141*x^14 - 476*x^13 + 7718*x^12 - 21126*x^11 + 204969*x^10 - 440625*x^9 + 2836720*x^8 - 4701297*x^7 + 19646115*x^6 - 24590405*x^5 + 54351103*x^4 - 58934929*x^3 - 17975866*x^2 - 15162339*x + 71134267)
 
gp: K = bnfinit(x^16 - 4*x^15 + 141*x^14 - 476*x^13 + 7718*x^12 - 21126*x^11 + 204969*x^10 - 440625*x^9 + 2836720*x^8 - 4701297*x^7 + 19646115*x^6 - 24590405*x^5 + 54351103*x^4 - 58934929*x^3 - 17975866*x^2 - 15162339*x + 71134267, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 141 x^{14} - 476 x^{13} + 7718 x^{12} - 21126 x^{11} + 204969 x^{10} - 440625 x^{9} + 2836720 x^{8} - 4701297 x^{7} + 19646115 x^{6} - 24590405 x^{5} + 54351103 x^{4} - 58934929 x^{3} - 17975866 x^{2} - 15162339 x + 71134267 \)

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:  \(26252865470156848284984700456389559681=13^{12}\cdot 101^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $218.12$
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:  $13, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{14020260} a^{12} - \frac{1}{4673420} a^{11} - \frac{2453173}{14020260} a^{10} + \frac{1869859}{14020260} a^{9} - \frac{230743}{1168355} a^{8} + \frac{1775347}{14020260} a^{7} - \frac{49765}{233671} a^{6} - \frac{5112311}{14020260} a^{5} - \frac{200075}{467342} a^{4} - \frac{4872143}{14020260} a^{3} + \frac{193157}{14020260} a^{2} + \frac{1143214}{3505065} a + \frac{5017}{77460}$, $\frac{1}{14020260} a^{13} - \frac{1226591}{7010130} a^{11} + \frac{152047}{1402026} a^{10} + \frac{946887}{4673420} a^{9} + \frac{478729}{14020260} a^{8} + \frac{780047}{4673420} a^{7} + \frac{6960379}{14020260} a^{6} + \frac{2233779}{4673420} a^{5} + \frac{5161627}{14020260} a^{4} + \frac{3303559}{7010130} a^{3} + \frac{5152327}{14020260} a^{2} - \frac{1280749}{2804052} a - \frac{7893}{25820}$, $\frac{1}{14020260} a^{14} + \frac{585527}{7010130} a^{11} + \frac{85393}{2804052} a^{10} - \frac{1685953}{14020260} a^{9} + \frac{728813}{4673420} a^{8} - \frac{4311997}{14020260} a^{7} - \frac{501431}{4673420} a^{6} - \frac{31277}{2804052} a^{5} - \frac{560158}{3505065} a^{4} - \frac{4747219}{14020260} a^{3} + \frac{48203}{4673420} a^{2} + \frac{6097963}{14020260} a + \frac{8356}{19365}$, $\frac{1}{6042495953327112237692276657675226660} a^{15} - \frac{31991508735295514239000883132}{1510623988331778059423069164418806665} a^{14} + \frac{10958993618958542689590759684}{503541329443926019807689721472935555} a^{13} - \frac{3054797558059243009950565817}{3021247976663556118846138328837613330} a^{12} + \frac{137264514525006237214379813386807177}{2014165317775704079230758885891742220} a^{11} - \frac{399078355833204461383579453446632263}{2014165317775704079230758885891742220} a^{10} - \frac{226151418709619923839354928999513127}{2014165317775704079230758885891742220} a^{9} - \frac{1245758517462109050193929189557050579}{6042495953327112237692276657675226660} a^{8} - \frac{139467904709975280961537315045973773}{402833063555140815846151777178348444} a^{7} + \frac{1591690571065401527689262373910997171}{6042495953327112237692276657675226660} a^{6} + \frac{697904006522964492180058218923902441}{3021247976663556118846138328837613330} a^{5} + \frac{86409531612415398609566329578764533}{6042495953327112237692276657675226660} a^{4} - \frac{15642577995848302325776095526332351}{118480312810335534072397581523043660} a^{3} - \frac{648962637201182551692761562297301533}{2014165317775704079230758885891742220} a^{2} + \frac{356018052294042833490367360908076667}{1007082658887852039615379442945871110} a + \frac{1084476337903501235896760674621337}{3338395554324371402039931855069186}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{10}\times C_{20}$, which has order $5000$ (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:  \( 151720134.33 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{101}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{1313}) \), 4.0.1030301.1, \(\Q(\sqrt{13}, \sqrt{101})\), 4.0.174120869.2, 4.0.221897.1 x2, 4.0.22411597.1 x2, 8.0.30318077021315161.5, 8.0.502279680090409.3, 8.8.5123755016602262209.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$101$101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$