↑

Properties

Label 16.0.17769124438...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{12}\cdot 11^{4}\cdot 41^{4}$
Root discriminant $103.66$
Ramified primes $2, 5, 11, 41$
Class number $348544$ (GRH)
Class group $[2, 2, 2, 2, 2, 10892]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T203)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6648587356, 6112206352, 5894282792, 2029216704, 1339583508, 234650416, 151448504, 11990232, 10154778, 191104, 430056, -7312, 11378, -352, 168, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 168*x^14 - 352*x^13 + 11378*x^12 - 7312*x^11 + 430056*x^10 + 191104*x^9 + 10154778*x^8 + 11990232*x^7 + 151448504*x^6 + 234650416*x^5 + 1339583508*x^4 + 2029216704*x^3 + 5894282792*x^2 + 6112206352*x + 6648587356)
 
gp: K = bnfinit(x^16 - 4*x^15 + 168*x^14 - 352*x^13 + 11378*x^12 - 7312*x^11 + 430056*x^10 + 191104*x^9 + 10154778*x^8 + 11990232*x^7 + 151448504*x^6 + 234650416*x^5 + 1339583508*x^4 + 2029216704*x^3 + 5894282792*x^2 + 6112206352*x + 6648587356, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 168 x^{14} - 352 x^{13} + 11378 x^{12} - 7312 x^{11} + 430056 x^{10} + 191104 x^{9} + 10154778 x^{8} + 11990232 x^{7} + 151448504 x^{6} + 234650416 x^{5} + 1339583508 x^{4} + 2029216704 x^{3} + 5894282792 x^{2} + 6112206352 x + 6648587356 \)

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:  \(177691244381492740096000000000000=2^{44}\cdot 5^{12}\cdot 11^{4}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $103.66$
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:  $2, 5, 11, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{22} a^{14} - \frac{1}{22} a^{12} - \frac{2}{11} a^{11} + \frac{3}{22} a^{10} - \frac{1}{11} a^{9} + \frac{1}{11} a^{8} + \frac{3}{11} a^{7} - \frac{5}{11} a^{6} - \frac{4}{11} a^{5} + \frac{3}{11} a^{4} - \frac{5}{11} a^{3} - \frac{5}{11} a^{2} - \frac{1}{11} a - \frac{5}{11}$, $\frac{1}{25652718995472472746354390334069273944893811495557278842980878} a^{15} - \frac{1813140517150060276568143006226059364971755197124935522184}{1166032681612385124834290469730421542949718704343512674680949} a^{14} + \frac{2806606601642800837457845112477895589469905410219328170832087}{25652718995472472746354390334069273944893811495557278842980878} a^{13} + \frac{953546591660854402432653037930227185703830433864264479778018}{12826359497736236373177195167034636972446905747778639421490439} a^{12} + \frac{1329185845977759992666956389294924613977879906924783472126165}{12826359497736236373177195167034636972446905747778639421490439} a^{11} + \frac{1510003553909177030999141806337432717918250825738139788518048}{12826359497736236373177195167034636972446905747778639421490439} a^{10} + \frac{1443994223433092540979585258613188192884511966932456095960917}{12826359497736236373177195167034636972446905747778639421490439} a^{9} + \frac{867173178506718727295283097031240089675175955676878191311199}{12826359497736236373177195167034636972446905747778639421490439} a^{8} + \frac{5694500040933420683969136904019194547877023593197795720278610}{12826359497736236373177195167034636972446905747778639421490439} a^{7} - \frac{3589014668835504196548042154982006396574639386647148462098893}{12826359497736236373177195167034636972446905747778639421490439} a^{6} - \frac{3687206050795525626969598476727955223644705914710154438896312}{12826359497736236373177195167034636972446905747778639421490439} a^{5} - \frac{2227517652303280284405451602129403127737557348167443576246344}{12826359497736236373177195167034636972446905747778639421490439} a^{4} - \frac{4589912305848303188050950121391229618567931265448317296195660}{12826359497736236373177195167034636972446905747778639421490439} a^{3} - \frac{4716004531071149867855138335633804775047505252886249748827377}{12826359497736236373177195167034636972446905747778639421490439} a^{2} + \frac{4856340501675650534618842060961091727039907694561617270479880}{12826359497736236373177195167034636972446905747778639421490439} a - \frac{376145788470984649928350923892868036938060042976071763450238}{1166032681612385124834290469730421542949718704343512674680949}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{10892}$, which has order $348544$ (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:  \( 7114.13535725 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T203):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 41 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{20})^+\), 4.4.8000.1, \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
$11$11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41Data not computed