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Properties

Label 16.0.40744439184...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{12}\cdot 5^{12}\cdot 13^{4}$
Root discriminant $81.87$
Ramified primes $2, 3, 5, 13$
Class number $28800$ (GRH)
Class group $[2, 2, 2, 2, 2, 30, 30]$ (GRH)
Galois group $(C_2 \times Q_8):C_2$ (as 16T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![246399481, 33597560, 73950516, 95409688, 5432442, 21942216, 9946672, -1786640, 2701602, -535048, 243628, -34032, 9606, -856, 168, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 168*x^14 - 856*x^13 + 9606*x^12 - 34032*x^11 + 243628*x^10 - 535048*x^9 + 2701602*x^8 - 1786640*x^7 + 9946672*x^6 + 21942216*x^5 + 5432442*x^4 + 95409688*x^3 + 73950516*x^2 + 33597560*x + 246399481)
 
gp: K = bnfinit(x^16 - 8*x^15 + 168*x^14 - 856*x^13 + 9606*x^12 - 34032*x^11 + 243628*x^10 - 535048*x^9 + 2701602*x^8 - 1786640*x^7 + 9946672*x^6 + 21942216*x^5 + 5432442*x^4 + 95409688*x^3 + 73950516*x^2 + 33597560*x + 246399481, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 168 x^{14} - 856 x^{13} + 9606 x^{12} - 34032 x^{11} + 243628 x^{10} - 535048 x^{9} + 2701602 x^{8} - 1786640 x^{7} + 9946672 x^{6} + 21942216 x^{5} + 5432442 x^{4} + 95409688 x^{3} + 73950516 x^{2} + 33597560 x + 246399481 \)

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:  \(4074443918442233856000000000000=2^{40}\cdot 3^{12}\cdot 5^{12}\cdot 13^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.87$
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:  $2, 3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{70} a^{12} + \frac{1}{5} a^{11} - \frac{4}{35} a^{10} - \frac{1}{7} a^{9} - \frac{8}{35} a^{8} - \frac{16}{35} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{9}{70} a^{4} - \frac{12}{35} a^{3} - \frac{1}{5} a^{2} - \frac{1}{35} a - \frac{2}{35}$, $\frac{1}{70} a^{13} + \frac{3}{35} a^{11} - \frac{3}{70} a^{10} - \frac{8}{35} a^{9} + \frac{17}{70} a^{8} + \frac{1}{5} a^{7} + \frac{1}{14} a^{5} + \frac{16}{35} a^{4} - \frac{2}{5} a^{3} + \frac{19}{70} a^{2} + \frac{12}{35} a + \frac{3}{10}$, $\frac{1}{70} a^{14} - \frac{17}{70} a^{11} - \frac{3}{70} a^{10} + \frac{1}{10} a^{9} + \frac{1}{14} a^{8} - \frac{9}{35} a^{7} + \frac{19}{70} a^{6} + \frac{9}{35} a^{5} + \frac{13}{35} a^{4} + \frac{23}{70} a^{3} + \frac{3}{70} a^{2} + \frac{33}{70} a - \frac{11}{70}$, $\frac{1}{15144531222874667417472525667690242655388456868966539039350} a^{15} + \frac{5629924242591241187685111337665795680792882872705522177}{1514453122287466741747252566769024265538845686896653903935} a^{14} - \frac{27392335298649041785925990247848191295714845957893350801}{7572265611437333708736262833845121327694228434483269519675} a^{13} + \frac{82384652531773349158274559883171688346161482519957170553}{15144531222874667417472525667690242655388456868966539039350} a^{12} - \frac{575720135662188825031165173492601755746868484814698748117}{3028906244574933483494505133538048531077691373793307807870} a^{11} - \frac{112653392255837703612180274583026296213952327720054744387}{15144531222874667417472525667690242655388456868966539039350} a^{10} + \frac{1372028643392662061099680106585126498097031713475368777931}{7572265611437333708736262833845121327694228434483269519675} a^{9} + \frac{1403730858068770230128611388768195260661513703234990643213}{15144531222874667417472525667690242655388456868966539039350} a^{8} + \frac{257748784355218350162372918052145073630223839663965803403}{2163504460410666773924646523955748950769779552709505577050} a^{7} + \frac{2284531295745490782260740933370185301744133169059268466024}{7572265611437333708736262833845121327694228434483269519675} a^{6} + \frac{245509288700673180768581867291379176982858486938379073314}{1081752230205333386962323261977874475384889776354752788525} a^{5} - \frac{6230229944513942301808354388396312953563532578117657365911}{15144531222874667417472525667690242655388456868966539039350} a^{4} + \frac{840789425893413035455771340368467632883358310718246372209}{15144531222874667417472525667690242655388456868966539039350} a^{3} + \frac{658144474176256988639106553709464599660055227216141043201}{3028906244574933483494505133538048531077691373793307807870} a^{2} + \frac{3657706982697914569258468740144620257317858607217526725863}{7572265611437333708736262833845121327694228434483269519675} a - \frac{7113317394142119964367202819326649437924081412277354461867}{15144531222874667417472525667690242655388456868966539039350}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{30}\times C_{30}$, which has order $28800$ (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:  \( 15197.4244561 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.C_2^3$ (as 16T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 17 conjugacy class representatives for $(C_2 \times Q_8):C_2$
Character table for $(C_2 \times Q_8):C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{5}, \sqrt{6})\), 8.8.3317760000.1

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$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}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$