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Properties

Label 16.8.42446930611...1209.3
Degree $16$
Signature $[8, 4]$
Discriminant $61^{14}\cdot 73^{10}$
Root discriminant $533.02$
Ramified primes $61, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6612908543775, -1974895242930, -3126568524328, 119599276971, 301450266691, 8803629494, -12091496560, -1022153529, 235889409, 32057398, -1807538, -373907, -13589, 2673, 55, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 55*x^14 + 2673*x^13 - 13589*x^12 - 373907*x^11 - 1807538*x^10 + 32057398*x^9 + 235889409*x^8 - 1022153529*x^7 - 12091496560*x^6 + 8803629494*x^5 + 301450266691*x^4 + 119599276971*x^3 - 3126568524328*x^2 - 1974895242930*x + 6612908543775)
 
gp: K = bnfinit(x^16 - 7*x^15 + 55*x^14 + 2673*x^13 - 13589*x^12 - 373907*x^11 - 1807538*x^10 + 32057398*x^9 + 235889409*x^8 - 1022153529*x^7 - 12091496560*x^6 + 8803629494*x^5 + 301450266691*x^4 + 119599276971*x^3 - 3126568524328*x^2 - 1974895242930*x + 6612908543775, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 55 x^{14} + 2673 x^{13} - 13589 x^{12} - 373907 x^{11} - 1807538 x^{10} + 32057398 x^{9} + 235889409 x^{8} - 1022153529 x^{7} - 12091496560 x^{6} + 8803629494 x^{5} + 301450266691 x^{4} + 119599276971 x^{3} - 3126568524328 x^{2} - 1974895242930 x + 6612908543775 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(42446930611030025911778708127377056380611209=61^{14}\cdot 73^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $533.02$
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:  $61, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{3} a^{6} - \frac{4}{9} a^{5} - \frac{1}{9} a^{4} + \frac{4}{9} a^{3} + \frac{2}{9} a^{2}$, $\frac{1}{27} a^{11} - \frac{1}{9} a^{9} - \frac{2}{27} a^{8} + \frac{4}{27} a^{7} + \frac{5}{27} a^{6} + \frac{2}{9} a^{5} - \frac{13}{27} a^{4} + \frac{10}{27} a^{3} + \frac{1}{27} a^{2} - \frac{1}{3}$, $\frac{1}{81} a^{12} + \frac{1}{81} a^{11} - \frac{11}{81} a^{9} - \frac{13}{81} a^{8} - \frac{4}{27} a^{7} - \frac{7}{81} a^{6} + \frac{8}{81} a^{5} + \frac{13}{27} a^{4} + \frac{14}{81} a^{3} + \frac{16}{81} a^{2} - \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{2158569} a^{13} + \frac{334}{79947} a^{12} + \frac{29339}{2158569} a^{11} - \frac{33059}{2158569} a^{10} + \frac{70765}{2158569} a^{9} + \frac{176437}{2158569} a^{8} - \frac{245218}{2158569} a^{7} + \frac{46643}{102789} a^{6} + \frac{898348}{2158569} a^{5} - \frac{11782}{308367} a^{4} + \frac{754643}{2158569} a^{3} + \frac{81710}{308367} a^{2} + \frac{89036}{239841} a + \frac{100540}{239841}$, $\frac{1}{2158569} a^{14} + \frac{11114}{2158569} a^{12} + \frac{12409}{2158569} a^{11} + \frac{74464}{2158569} a^{10} - \frac{245114}{2158569} a^{9} + \frac{310739}{2158569} a^{8} + \frac{47330}{719523} a^{7} - \frac{194828}{2158569} a^{6} - \frac{275281}{2158569} a^{5} - \frac{41236}{2158569} a^{4} + \frac{769718}{2158569} a^{3} - \frac{35872}{239841} a^{2} + \frac{11659}{34263} a + \frac{3302}{8883}$, $\frac{1}{15701821307511759504720239878109741024193910560985984981015944033284808015} a^{15} + \frac{90666347663639618513532602240534403933702158467387734798268115383}{747705776548179024034297137052844810675900502904094522905521144442133715} a^{14} + \frac{348443082586287911161792108898971629813618342081535698030950034833}{3140364261502351900944047975621948204838782112197196996203188806656961603} a^{13} - \frac{4443193105436545801487793939549150586287887588782308528160972373674661}{2243117329644537072102891411158534432027701508712283568716563433326401145} a^{12} + \frac{4691777043867907246197814919190701928168816266803078314804722093195798}{2243117329644537072102891411158534432027701508712283568716563433326401145} a^{11} + \frac{5563993942622241691374539711571024259035626962506693350064931769212003}{747705776548179024034297137052844810675900502904094522905521144442133715} a^{10} + \frac{1981369484953422337232459313335986618026486862626869277925581829362480532}{15701821307511759504720239878109741024193910560985984981015944033284808015} a^{9} + \frac{335007291091392220636370381523728790036599030007722011467882804775884098}{15701821307511759504720239878109741024193910560985984981015944033284808015} a^{8} + \frac{1322539368835347758024784494616020891390728274746478594680710543053697044}{15701821307511759504720239878109741024193910560985984981015944033284808015} a^{7} + \frac{2973490960381178719034060104215514006243674138058076977476593796461290301}{15701821307511759504720239878109741024193910560985984981015944033284808015} a^{6} - \frac{625221660089680341524006863030057577011384425612208002846921080105558074}{3140364261502351900944047975621948204838782112197196996203188806656961603} a^{5} + \frac{808539020767105705297786871059322114827164549027246398664799363173790406}{1744646811945751056080026653123304558243767840109553886779549337031645335} a^{4} + \frac{163644680965918287836669452854218340244912795262076542700707188681533178}{2243117329644537072102891411158534432027701508712283568716563433326401145} a^{3} - \frac{890218768398675106274610815010994557437411051747229339401226841202363297}{2243117329644537072102891411158534432027701508712283568716563433326401145} a^{2} - \frac{544682150841325218558029964587109944382925258895777805217412856731784887}{1744646811945751056080026653123304558243767840109553886779549337031645335} a - \frac{157572734478456262664037424664950503329841451064771312897528808422311550}{348929362389150211216005330624660911648753568021910777355909867406329067}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 262280316918000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{4453}) \), \(\Q(\sqrt{73}) \), 4.4.16569613.1 x2, 4.4.1209581749.1 x2, \(\Q(\sqrt{61}, \sqrt{73})\), 8.8.1463088007513899001.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$61$61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$73$73.8.4.1$x^{8} + 138554 x^{4} - 389017 x^{2} + 4799302729$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
73.8.6.2$x^{8} + 1533 x^{4} + 644809$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$