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Properties

Label 16.0.80341646997...4929.2
Degree $16$
Signature $[0, 8]$
Discriminant $37^{12}\cdot 73^{14}$
Root discriminant $640.56$
Ramified primes $37, 73$
Class number $2624$ (GRH)
Class group $[2, 2, 2, 328]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51165489031, -20284389474, 23803505264, -7391392826, 4600537631, -1148053948, 488894886, -96026660, 30357096, -4411234, 1060966, -98908, 17698, -588, 104, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 104*x^14 - 588*x^13 + 17698*x^12 - 98908*x^11 + 1060966*x^10 - 4411234*x^9 + 30357096*x^8 - 96026660*x^7 + 488894886*x^6 - 1148053948*x^5 + 4600537631*x^4 - 7391392826*x^3 + 23803505264*x^2 - 20284389474*x + 51165489031)
 
gp: K = bnfinit(x^16 - 8*x^15 + 104*x^14 - 588*x^13 + 17698*x^12 - 98908*x^11 + 1060966*x^10 - 4411234*x^9 + 30357096*x^8 - 96026660*x^7 + 488894886*x^6 - 1148053948*x^5 + 4600537631*x^4 - 7391392826*x^3 + 23803505264*x^2 - 20284389474*x + 51165489031, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 104 x^{14} - 588 x^{13} + 17698 x^{12} - 98908 x^{11} + 1060966 x^{10} - 4411234 x^{9} + 30357096 x^{8} - 96026660 x^{7} + 488894886 x^{6} - 1148053948 x^{5} + 4600537631 x^{4} - 7391392826 x^{3} + 23803505264 x^{2} - 20284389474 x + 51165489031 \)

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:  \(803416469975725073264940954221144185207724929=37^{12}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $640.56$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{74} a^{4} - \frac{1}{37} a^{3} + \frac{10}{37} a^{2} - \frac{19}{74} a + \frac{7}{74}$, $\frac{1}{74} a^{5} + \frac{8}{37} a^{3} + \frac{21}{74} a^{2} - \frac{31}{74} a + \frac{7}{37}$, $\frac{1}{74} a^{6} - \frac{21}{74} a^{3} + \frac{19}{74} a^{2} + \frac{11}{37} a + \frac{18}{37}$, $\frac{1}{74} a^{7} - \frac{23}{74} a^{3} - \frac{1}{37} a^{2} + \frac{7}{74} a - \frac{1}{74}$, $\frac{1}{32856} a^{8} - \frac{1}{8214} a^{7} + \frac{11}{8214} a^{6} - \frac{59}{16428} a^{5} - \frac{7}{8214} a^{4} + \frac{31}{4107} a^{3} - \frac{9719}{32856} a^{2} + \frac{399}{1369} a + \frac{7375}{32856}$, $\frac{1}{32856} a^{9} + \frac{7}{8214} a^{7} + \frac{29}{16428} a^{6} - \frac{7}{4107} a^{5} + \frac{17}{4107} a^{4} - \frac{541}{10952} a^{3} + \frac{1610}{4107} a^{2} - \frac{941}{32856} a + \frac{715}{8214}$, $\frac{1}{32856} a^{10} + \frac{85}{16428} a^{7} + \frac{11}{8214} a^{6} - \frac{14}{4107} a^{5} + \frac{49}{32856} a^{4} - \frac{3733}{8214} a^{3} + \frac{3225}{10952} a^{2} - \frac{2825}{8214} a - \frac{1231}{8214}$, $\frac{1}{32856} a^{11} - \frac{41}{8214} a^{7} - \frac{11}{8214} a^{6} + \frac{43}{10952} a^{5} + \frac{5}{4107} a^{4} + \frac{15023}{32856} a^{3} - \frac{3155}{16428} a^{2} + \frac{935}{8214} a - \frac{3277}{16428}$, $\frac{1}{77803008} a^{12} - \frac{1}{12967168} a^{11} + \frac{3}{3241792} a^{10} - \frac{305}{77803008} a^{9} - \frac{383}{77803008} a^{8} + \frac{103}{2431344} a^{7} - \frac{76601}{77803008} a^{6} + \frac{54263}{19450752} a^{5} - \frac{22995}{25934336} a^{4} - \frac{219361}{77803008} a^{3} + \frac{5657403}{25934336} a^{2} - \frac{16826989}{77803008} a + \frac{20888323}{77803008}$, $\frac{1}{77803008} a^{13} + \frac{3}{6483584} a^{11} + \frac{127}{77803008} a^{10} + \frac{155}{77803008} a^{9} + \frac{499}{38901504} a^{8} + \frac{9479}{77803008} a^{7} - \frac{17535}{12967168} a^{6} + \frac{49327}{77803008} a^{5} - \frac{103741}{25934336} a^{4} - \frac{1669831}{25934336} a^{3} + \frac{5207955}{25934336} a^{2} + \frac{28094261}{77803008} a - \frac{19111543}{38901504}$, $\frac{1}{77803008} a^{14} + \frac{343}{77803008} a^{11} - \frac{23}{25934336} a^{10} + \frac{23}{12967168} a^{9} - \frac{413}{77803008} a^{8} - \frac{29053}{38901504} a^{7} + \frac{131123}{77803008} a^{6} + \frac{323929}{77803008} a^{5} - \frac{67675}{25934336} a^{4} + \frac{7861599}{25934336} a^{3} - \frac{17632879}{77803008} a^{2} + \frac{223857}{12967168} a + \frac{4830149}{19450752}$, $\frac{1}{3130870844928} a^{15} + \frac{20113}{3130870844928} a^{14} - \frac{415}{130452951872} a^{13} - \frac{29}{16306618984} a^{12} - \frac{2471093}{782717711232} a^{11} + \frac{39827621}{3130870844928} a^{10} + \frac{552405}{260905903744} a^{9} - \frac{16846007}{1565435422464} a^{8} - \frac{16820104031}{3130870844928} a^{7} + \frac{9965123819}{3130870844928} a^{6} - \frac{1248681073}{782717711232} a^{5} + \frac{6072986611}{3130870844928} a^{4} - \frac{171901358465}{1043623614976} a^{3} + \frac{4833102981}{130452951872} a^{2} + \frac{944181507677}{3130870844928} a - \frac{204083628651}{1043623614976}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{328}$, which has order $2624$ (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:  \( 1782108537960 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.0.20704603455949352617.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
37Data not computed
$73$73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$