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Properties

Label 16.0.13742275226...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{30}\cdot 5^{10}\cdot 41^{6}\cdot 359^{8}$
Root discriminant $764.94$
Ramified primes $2, 5, 41, 359$
Class number $6912$ (GRH)
Class group $[2, 2, 2, 6, 144]$ (GRH)
Galois group 16T1220

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13594580577640000, 0, 178926406536000, 0, 2284011012880, 0, 51447741360, 0, 517543324, 0, 3312588, 0, 46558, 0, 234, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 234*x^14 + 46558*x^12 + 3312588*x^10 + 517543324*x^8 + 51447741360*x^6 + 2284011012880*x^4 + 178926406536000*x^2 + 13594580577640000)
 
gp: K = bnfinit(x^16 + 234*x^14 + 46558*x^12 + 3312588*x^10 + 517543324*x^8 + 51447741360*x^6 + 2284011012880*x^4 + 178926406536000*x^2 + 13594580577640000, 1)
 

Normalized defining polynomial

\( x^{16} + 234 x^{14} + 46558 x^{12} + 3312588 x^{10} + 517543324 x^{8} + 51447741360 x^{6} + 2284011012880 x^{4} + 178926406536000 x^{2} + 13594580577640000 \)

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:  \(13742275226577042645713355852595855360000000000=2^{30}\cdot 5^{10}\cdot 41^{6}\cdot 359^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $764.94$
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:  $2, 5, 41, 359$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{6560} a^{12} - \frac{1}{16} a^{11} - \frac{11}{410} a^{10} - \frac{91}{3280} a^{8} + \frac{1}{8} a^{7} - \frac{129}{820} a^{6} + \frac{81}{1640} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6560} a^{13} - \frac{11}{410} a^{11} - \frac{91}{3280} a^{9} - \frac{129}{820} a^{7} + \frac{81}{1640} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{1274356527064955054852781209867429220001374400} a^{14} - \frac{22831069307955203519420937762705225500539}{318589131766238763713195302466857305000343600} a^{12} - \frac{19902363013899901309095301354425074448798351}{637178263532477527426390604933714610000687200} a^{10} - \frac{1}{8} a^{9} - \frac{6694568205555408524757453236467148557324877}{79647282941559690928298825616714326250085900} a^{8} - \frac{1}{4} a^{7} - \frac{23970079891509404334354241229291888650639299}{318589131766238763713195302466857305000343600} a^{6} - \frac{1}{4} a^{5} + \frac{617741783947552797974754178907685635149107}{6371782635324775274263906049337146100006872} a^{4} - \frac{1}{2} a^{3} - \frac{33404166537892774465255630428814645885069}{388523331422242394772189393252265006097980} a^{2} - \frac{1}{2} a - \frac{1559923326005984585618919962670952468568}{19426166571112119738609469662613250304899}$, $\frac{1}{90600377291682979624758480115524880395997712968000} a^{15} - \frac{1206582036801081712168553581682671681662777429}{22650094322920744906189620028881220098999428242000} a^{13} - \frac{446015159062560078677779832361067479934466391871}{45300188645841489812379240057762440197998856484000} a^{11} + \frac{4303228791765519298499866770656329318487290887}{34527582809330403820411006141587225760669860125} a^{9} + \frac{4697058372922496012616052001875915671473342017181}{22650094322920744906189620028881220098999428242000} a^{7} + \frac{162545631938062781592404073894033475926325438739}{2265009432292074490618962002888122009899942824200} a^{5} + \frac{8051335592896591254089615540736258906722554471}{27622066247464323056328804913269780608535888100} a^{3} + \frac{127520815854743197615210878869265886110985282}{276220662474643230563288049132697806085358881} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{144}$, which has order $6912$ (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:  \( 14585549456700 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1220:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1220 are not computed
Character table for t16n1220 is not computed

Intermediate fields

\(\Q(\sqrt{359}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{1795}) \), 4.0.1025.1, 4.0.2113648400.1, \(\Q(\sqrt{5}, \sqrt{359})\), 8.0.4467509558822560000.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.8.12.20$x^{8} + 8 x^{6} + 12 x^{4} + 80$$4$$2$$12$$C_2^3: C_4$$[2, 2, 2]^{4}$
2.8.18.19$x^{8} + 16 x^{5} + 36$$4$$2$$18$$C_2^3: C_4$$[2, 2, 3, 7/2]^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
359Data not computed