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Properties

Label 16.0.45137758519...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{16}\cdot 5^{12}$
Root discriminant $40.12$
Ramified primes $2, 3, 5$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group $A_4:C_4$ (as 16T62)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1296, 4320, 11232, 9600, 5608, -19920, 42144, -34720, 22020, -5520, 1824, -240, 138, -20, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 12*x^14 - 20*x^13 + 138*x^12 - 240*x^11 + 1824*x^10 - 5520*x^9 + 22020*x^8 - 34720*x^7 + 42144*x^6 - 19920*x^5 + 5608*x^4 + 9600*x^3 + 11232*x^2 + 4320*x + 1296)
 
gp: K = bnfinit(x^16 + 12*x^14 - 20*x^13 + 138*x^12 - 240*x^11 + 1824*x^10 - 5520*x^9 + 22020*x^8 - 34720*x^7 + 42144*x^6 - 19920*x^5 + 5608*x^4 + 9600*x^3 + 11232*x^2 + 4320*x + 1296, 1)
 

Normalized defining polynomial

\( x^{16} + 12 x^{14} - 20 x^{13} + 138 x^{12} - 240 x^{11} + 1824 x^{10} - 5520 x^{9} + 22020 x^{8} - 34720 x^{7} + 42144 x^{6} - 19920 x^{5} + 5608 x^{4} + 9600 x^{3} + 11232 x^{2} + 4320 x + 1296 \)

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:  \(45137758519296000000000000=2^{32}\cdot 3^{16}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.12$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{20} a^{8} - \frac{1}{5} a^{6} - \frac{1}{5} a^{4} - \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{60} a^{9} - \frac{1}{60} a^{8} - \frac{7}{30} a^{7} + \frac{1}{15} a^{6} + \frac{1}{10} a^{5} + \frac{7}{30} a^{4} + \frac{4}{15} a^{3} + \frac{1}{15} a^{2} + \frac{4}{15} a + \frac{2}{5}$, $\frac{1}{60} a^{10} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{2}{5}$, $\frac{1}{60} a^{11} - \frac{1}{60} a^{8} + \frac{1}{6} a^{7} + \frac{7}{30} a^{6} + \frac{7}{30} a^{4} + \frac{1}{3} a^{3} + \frac{1}{15} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{120} a^{12} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{7}{15} a^{2} + \frac{1}{3} a$, $\frac{1}{120} a^{13} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{7}{15} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{4544225640} a^{14} - \frac{345113}{757370940} a^{13} - \frac{959509}{504913960} a^{12} - \frac{4059209}{1136056410} a^{11} - \frac{106771}{189342735} a^{10} - \frac{3816463}{757370940} a^{9} - \frac{2775623}{378685470} a^{8} + \frac{24370963}{189342735} a^{7} - \frac{38235094}{189342735} a^{6} - \frac{16959487}{1136056410} a^{5} - \frac{34273573}{189342735} a^{4} - \frac{15066290}{37868547} a^{3} - \frac{31619471}{113605641} a^{2} - \frac{90012583}{189342735} a - \frac{5730701}{63114245}$, $\frac{1}{590105157405950520} a^{15} - \frac{421783}{32783619855886140} a^{14} - \frac{27080197387163}{19670171913531684} a^{13} - \frac{496148280494333}{147526289351487630} a^{12} - \frac{160406101462793}{49175429783829210} a^{11} + \frac{23366059748201}{98350859567658420} a^{10} - \frac{252364322517353}{98350859567658420} a^{9} + \frac{1083865217311501}{98350859567658420} a^{8} - \frac{2593209718316329}{49175429783829210} a^{7} + \frac{3432634629463402}{73763144675743815} a^{6} + \frac{1203728319018679}{24587714891914605} a^{5} + \frac{151052311216865}{4917542978382921} a^{4} + \frac{10807723427789339}{73763144675743815} a^{3} + \frac{11239197949391872}{24587714891914605} a^{2} - \frac{925484789788441}{8195904963971535} a + \frac{68992833838167}{546393664264769}$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  \( -\frac{36260866329866}{73763144675743815} a^{15} + \frac{3274332265007}{24587714891914605} a^{14} - \frac{288982085079439}{49175429783829210} a^{13} + \frac{846246983987788}{73763144675743815} a^{12} - \frac{575411138923772}{8195904963971535} a^{11} + \frac{3354012119281894}{24587714891914605} a^{10} - \frac{4554108499875152}{4917542978382921} a^{9} + \frac{58149740444607559}{19670171913531684} a^{8} - \frac{56678331504758456}{4917542978382921} a^{7} + \frac{294271547942267356}{14752628935148763} a^{6} - \frac{205187121197465716}{8195904963971535} a^{5} + \frac{378958486003969568}{24587714891914605} a^{4} - \frac{396086292722139344}{73763144675743815} a^{3} - \frac{88945045923931016}{24587714891914605} a^{2} - \frac{31924432445451848}{8195904963971535} a - \frac{1187265434453216}{2731968321323845} \) (order $10$)
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:  \( 740354.925982 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_4:C_4$ (as 16T62):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 10 conjugacy class representatives for $A_4:C_4$
Character table for $A_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2592000.1, \(\Q(\zeta_{5})\), 8.8.6718464000000.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 24 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.12.16.30$x^{12} + 93 x^{11} + 351 x^{10} + 3 x^{9} + 126 x^{8} - 297 x^{7} + 171 x^{6} + 243 x^{5} - 324 x^{4} - 54 x^{3} + 162 x^{2} - 243 x + 324$$3$$4$$16$$C_3 : C_4$$[2]^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$