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Properties

Label 16.0.78039659508...8064.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 3^{12}\cdot 17^{6}\cdot 97^{4}$
Root discriminant $98.46$
Ramified primes $2, 3, 17, 97$
Class number $180288$ (GRH)
Class group $[2, 2, 2, 2, 2, 5634]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T657)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1587808969, 1155456948, 1237648906, -153572468, 495625354, -122570424, 86507434, -18508272, 7667398, -1294564, 381910, -48692, 10922, -968, 166, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 166*x^14 - 968*x^13 + 10922*x^12 - 48692*x^11 + 381910*x^10 - 1294564*x^9 + 7667398*x^8 - 18508272*x^7 + 86507434*x^6 - 122570424*x^5 + 495625354*x^4 - 153572468*x^3 + 1237648906*x^2 + 1155456948*x + 1587808969)
 
gp: K = bnfinit(x^16 - 8*x^15 + 166*x^14 - 968*x^13 + 10922*x^12 - 48692*x^11 + 381910*x^10 - 1294564*x^9 + 7667398*x^8 - 18508272*x^7 + 86507434*x^6 - 122570424*x^5 + 495625354*x^4 - 153572468*x^3 + 1237648906*x^2 + 1155456948*x + 1587808969, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 166 x^{14} - 968 x^{13} + 10922 x^{12} - 48692 x^{11} + 381910 x^{10} - 1294564 x^{9} + 7667398 x^{8} - 18508272 x^{7} + 86507434 x^{6} - 122570424 x^{5} + 495625354 x^{4} - 153572468 x^{3} + 1237648906 x^{2} + 1155456948 x + 1587808969 \)

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:  \(78039659508312536877345552728064=2^{36}\cdot 3^{12}\cdot 17^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.46$
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:  $2, 3, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{14} a^{13} - \frac{3}{14} a^{12} - \frac{1}{7} a^{11} - \frac{3}{14} a^{9} - \frac{3}{14} a^{8} - \frac{2}{7} a^{6} - \frac{5}{14} a^{5} - \frac{1}{2} a^{4} + \frac{2}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{14} a - \frac{5}{14}$, $\frac{1}{1274} a^{14} - \frac{3}{91} a^{13} + \frac{138}{637} a^{12} - \frac{223}{1274} a^{11} - \frac{5}{637} a^{10} + \frac{317}{1274} a^{9} + \frac{201}{1274} a^{8} - \frac{289}{637} a^{7} + \frac{445}{1274} a^{6} + \frac{255}{637} a^{5} + \frac{12}{49} a^{4} + \frac{555}{1274} a^{3} + \frac{193}{637} a^{2} + \frac{627}{1274} a + \frac{503}{1274}$, $\frac{1}{1658653840101005327484281726543246693936730350171195719598} a^{15} + \frac{30036493880369300988766469820724862143210424257745187}{1658653840101005327484281726543246693936730350171195719598} a^{14} + \frac{1812169516290461866246372951060685449118296496574839026}{63794378465423281826318527943971026689874244237353681523} a^{13} + \frac{168764756925410249803798336909118320163746035476093798133}{829326920050502663742140863271623346968365175085597859799} a^{12} - \frac{2199278197810971760558441571256684887867284797787905423}{16925039184704135994737568638196394836089085205828527751} a^{11} + \frac{26124417309466695447111145686241320806108834676172441023}{236950548585857903926325960934749527705247192881599388514} a^{10} + \frac{303573817363712876507705316663932091879969900330572556109}{1658653840101005327484281726543246693936730350171195719598} a^{9} - \frac{11160686220930430988453968068814456265802569494637046630}{63794378465423281826318527943971026689874244237353681523} a^{8} - \frac{206101204450567706888098957015032120541731167578907709615}{1658653840101005327484281726543246693936730350171195719598} a^{7} - \frac{168266146548515247458505178072899768425446474168075137333}{1658653840101005327484281726543246693936730350171195719598} a^{6} - \frac{388265491368803516696424297711809271191619980318623301912}{829326920050502663742140863271623346968365175085597859799} a^{5} + \frac{51345305169267003873345799278391900754186434322057262331}{118475274292928951963162980467374763852623596440799694257} a^{4} + \frac{39621482428441235994387561218656307565490784477438913343}{118475274292928951963162980467374763852623596440799694257} a^{3} + \frac{5160644615053075028406062137755025809223493540752209919}{236950548585857903926325960934749527705247192881599388514} a^{2} - \frac{622335827752479001819345109218310654217552665000604737373}{1658653840101005327484281726543246693936730350171195719598} a - \frac{312701607541809270653444685691182192629819205391791442673}{829326920050502663742140863271623346968365175085597859799}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{5634}$, which has order $180288$ (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:  \( 9072.35800888 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T657):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), 4.4.4352.1, 4.4.9792.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.1534132224.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
3Data not computed
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97Data not computed