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Properties

Label 16.8.80291031759...9473.3
Degree $16$
Signature $[8, 4]$
Discriminant $37^{14}\cdot 73^{15}$
Root discriminant $1315.35$
Ramified primes $37, 73$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21318855083198401, -8572126998875658, 1279771953158790, 163951671539017, -34266716442, -595121748833, -744609210817, -14853519562, 5939864526, 67725940, 9329744, 234117, -145134, 119, -29, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 29*x^14 + 119*x^13 - 145134*x^12 + 234117*x^11 + 9329744*x^10 + 67725940*x^9 + 5939864526*x^8 - 14853519562*x^7 - 744609210817*x^6 - 595121748833*x^5 - 34266716442*x^4 + 163951671539017*x^3 + 1279771953158790*x^2 - 8572126998875658*x + 21318855083198401)
 
gp: K = bnfinit(x^16 - 4*x^15 - 29*x^14 + 119*x^13 - 145134*x^12 + 234117*x^11 + 9329744*x^10 + 67725940*x^9 + 5939864526*x^8 - 14853519562*x^7 - 744609210817*x^6 - 595121748833*x^5 - 34266716442*x^4 + 163951671539017*x^3 + 1279771953158790*x^2 - 8572126998875658*x + 21318855083198401, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 29 x^{14} + 119 x^{13} - 145134 x^{12} + 234117 x^{11} + 9329744 x^{10} + 67725940 x^{9} + 5939864526 x^{8} - 14853519562 x^{7} - 744609210817 x^{6} - 595121748833 x^{5} - 34266716442 x^{4} + 163951671539017 x^{3} + 1279771953158790 x^{2} - 8572126998875658 x + 21318855083198401 \)

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:  \(80291031759964036646878404141998486437104406229473=37^{14}\cdot 73^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1315.35$
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:  $37, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{2} a^{8} + \frac{1}{16} a^{7} + \frac{5}{16} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{3}{16} a^{3} + \frac{7}{16} a^{2} - \frac{1}{2} a - \frac{7}{16}$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} + \frac{3}{32} a^{9} - \frac{7}{64} a^{8} + \frac{11}{32} a^{7} + \frac{13}{64} a^{6} + \frac{3}{16} a^{5} - \frac{25}{64} a^{4} - \frac{3}{32} a^{3} - \frac{1}{64} a^{2} - \frac{31}{64} a + \frac{25}{64}$, $\frac{1}{256} a^{12} + \frac{5}{256} a^{10} - \frac{1}{256} a^{9} + \frac{15}{256} a^{8} - \frac{29}{256} a^{7} + \frac{25}{256} a^{6} + \frac{115}{256} a^{5} - \frac{31}{256} a^{4} + \frac{121}{256} a^{3} - \frac{3}{8} a^{2} + \frac{29}{128} a + \frac{89}{256}$, $\frac{1}{1024} a^{13} + \frac{1}{1024} a^{12} + \frac{5}{1024} a^{11} + \frac{1}{256} a^{10} + \frac{7}{512} a^{9} - \frac{7}{512} a^{8} + \frac{127}{256} a^{7} - \frac{93}{256} a^{6} + \frac{21}{256} a^{5} + \frac{45}{512} a^{4} - \frac{487}{1024} a^{3} - \frac{147}{512} a^{2} + \frac{403}{1024} a + \frac{89}{1024}$, $\frac{1}{162494658012196864} a^{14} + \frac{39085728789657}{81247329006098432} a^{13} + \frac{17554172168251}{81247329006098432} a^{12} + \frac{479401102289785}{162494658012196864} a^{11} - \frac{865809196151351}{81247329006098432} a^{10} + \frac{252516250748115}{5077958062881152} a^{9} - \frac{15370468344842809}{81247329006098432} a^{8} - \frac{9434220081553455}{20311832251524608} a^{7} + \frac{769108861194607}{5077958062881152} a^{6} + \frac{33679013617288215}{81247329006098432} a^{5} - \frac{45562485454425325}{162494658012196864} a^{4} + \frac{60099218757118691}{162494658012196864} a^{3} - \frac{50804846105027123}{162494658012196864} a^{2} - \frac{12224099859712353}{40623664503049216} a - \frac{57494748915182391}{162494658012196864}$, $\frac{1}{4898122987471580560614057025167213966281224758531700037732445438196489015109516379275264} a^{15} + \frac{392411648033765921543917741667074371002035752349581345741524626672035}{4898122987471580560614057025167213966281224758531700037732445438196489015109516379275264} a^{14} - \frac{178133875021508698446035929582441688484767923974339689766660215802278358256321557945}{612265373433947570076757128145901745785153094816462504716555679774561126888689547409408} a^{13} - \frac{934021053734055251389508643716441596520063567262640241736699395098570730206019580097}{4898122987471580560614057025167213966281224758531700037732445438196489015109516379275264} a^{12} - \frac{25332379556513285991468958482390884221836224373929640806153584211369209922363047282069}{4898122987471580560614057025167213966281224758531700037732445438196489015109516379275264} a^{11} - \frac{4688054581566915502157529666737182964037368036692255776626951030278461047228107175863}{2449061493735790280307028512583606983140612379265850018866222719098244507554758189637632} a^{10} + \frac{284220314963933246220438075697559844855160816357235998226238517278950816890429462243143}{2449061493735790280307028512583606983140612379265850018866222719098244507554758189637632} a^{9} - \frac{974500476298894688568568244654067220606400211906719837470688427805572587039389563181237}{2449061493735790280307028512583606983140612379265850018866222719098244507554758189637632} a^{8} + \frac{191724468765661143993290652726623321324161257465479247386864097928850062358433438625093}{612265373433947570076757128145901745785153094816462504716555679774561126888689547409408} a^{7} + \frac{162281727360154862456789601084731756467030273226550787053670563048609966923723642061095}{2449061493735790280307028512583606983140612379265850018866222719098244507554758189637632} a^{6} + \frac{866690608275088494906667787152043734055356529926443239421970873767355260861254841558049}{4898122987471580560614057025167213966281224758531700037732445438196489015109516379275264} a^{5} + \frac{1129235977715534086379604713739801813712105976858990417918604456736377685818480168736275}{2449061493735790280307028512583606983140612379265850018866222719098244507554758189637632} a^{4} - \frac{128423107300357341584818966564249460034741515736027939469699376162159734441831070718749}{306132686716973785038378564072950872892576547408231252358277839887280563444344773704704} a^{3} - \frac{994524867665917736044291053479337161166921196289162518933160147785648532080304079856935}{4898122987471580560614057025167213966281224758531700037732445438196489015109516379275264} a^{2} - \frac{174258168514444067722936770248328480461538920670637444582511469067512968960243601984619}{4898122987471580560614057025167213966281224758531700037732445438196489015109516379275264} a + \frac{2243362997350255145986736631189656601157244058577785909548085241045751367093038325787113}{4898122987471580560614057025167213966281224758531700037732445438196489015109516379275264}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( 828352466885000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.532564273.1, 8.8.28344602131194663732673.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ $16$ $16$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ $16$ $16$ $16$ $16$

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
$37$37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
73Data not computed