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Properties

Label 16.0.70367540303...0128.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{45}\cdot 7^{4}\cdot 97^{6}$
Root discriminant $63.53$
Ramified primes $2, 7, 97$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group 16T1433

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![61140096, 55615680, 34893252, 22638336, 11430398, 4107528, 1622334, 480168, 80197, 9896, -1332, -2200, -520, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^14 - 520*x^12 - 2200*x^11 - 1332*x^10 + 9896*x^9 + 80197*x^8 + 480168*x^7 + 1622334*x^6 + 4107528*x^5 + 11430398*x^4 + 22638336*x^3 + 34893252*x^2 + 55615680*x + 61140096)
 
gp: K = bnfinit(x^16 - 6*x^14 - 520*x^12 - 2200*x^11 - 1332*x^10 + 9896*x^9 + 80197*x^8 + 480168*x^7 + 1622334*x^6 + 4107528*x^5 + 11430398*x^4 + 22638336*x^3 + 34893252*x^2 + 55615680*x + 61140096, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{14} - 520 x^{12} - 2200 x^{11} - 1332 x^{10} + 9896 x^{9} + 80197 x^{8} + 480168 x^{7} + 1622334 x^{6} + 4107528 x^{5} + 11430398 x^{4} + 22638336 x^{3} + 34893252 x^{2} + 55615680 x + 61140096 \)

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:  \(70367540303366615290376880128=2^{45}\cdot 7^{4}\cdot 97^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.53$
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:  $2, 7, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{12} a^{10} - \frac{1}{3} a^{7} - \frac{1}{4} a^{6} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{24} a^{11} - \frac{1}{6} a^{8} + \frac{3}{8} a^{7} - \frac{5}{12} a^{5} - \frac{1}{6} a^{4} + \frac{5}{12} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{12} + \frac{1}{24} a^{8} - \frac{1}{3} a^{7} - \frac{5}{12} a^{6} + \frac{1}{3} a^{5} + \frac{5}{12} a^{4} + \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{24} a^{13} + \frac{1}{24} a^{9} + \frac{1}{6} a^{8} - \frac{5}{12} a^{7} + \frac{1}{3} a^{6} + \frac{5}{12} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{1224} a^{14} - \frac{1}{102} a^{13} - \frac{7}{408} a^{12} + \frac{1}{68} a^{11} - \frac{49}{1224} a^{10} - \frac{11}{153} a^{9} - \frac{15}{136} a^{8} + \frac{157}{612} a^{7} + \frac{89}{612} a^{6} - \frac{25}{51} a^{5} - \frac{79}{204} a^{4} + \frac{25}{102} a^{3} - \frac{47}{153} a^{2} + \frac{13}{51} a + \frac{3}{17}$, $\frac{1}{23388452557449446413057076025754162592610348912} a^{15} - \frac{14020188840230812779095283244660918973801}{487259428280196800272022417203211720679382269} a^{14} - \frac{1364613618196761637047149439156052165756807}{3898075426241574402176179337625693765435058152} a^{13} - \frac{2951689849070176439002508828928179191671145}{1299358475413858134058726445875231255145019384} a^{12} + \frac{7738362984871527359578281241845920972302148}{1461778284840590400816067251609635162038146807} a^{11} + \frac{69209745646467031769824006238200989825018253}{2923556569681180801632134503219270324076293614} a^{10} + \frac{28801495236750134862901685777039653576168661}{487259428280196800272022417203211720679382269} a^{9} + \frac{174369296578266600532396751136098710080787939}{11694226278724723206528538012877081296305174456} a^{8} - \frac{9301132228181209640673680281663227791767535771}{23388452557449446413057076025754162592610348912} a^{7} - \frac{461158967010108718955195865044981239423122395}{1949037713120787201088089668812846882717529076} a^{6} - \frac{607424578562458740516146219020333015062323107}{3898075426241574402176179337625693765435058152} a^{5} - \frac{2770819539155489807471157409696851799577003}{38216425747466415707609601349271507504265276} a^{4} + \frac{4410228112075299813874050845472484679091070975}{11694226278724723206528538012877081296305174456} a^{3} + \frac{18003853282301056374652684325826608068439419}{57324638621199623561414402023907261256397914} a^{2} + \frac{21553240191840402080709133441936922266057319}{216559745902309689009787740979205209190836564} a + \frac{14574891654871135945323794698050120847745234}{54139936475577422252446935244801302297709141}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( 75243551.1838 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1433:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 68 conjugacy class representatives for t16n1433 are not computed
Character table for t16n1433 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.6208.2, 8.0.1233256448.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $16$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ $16$

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
$2$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.8.28.20$x^{8} + 8 x^{7} + 8 x^{5} + 30$$8$$1$$28$$C_2^3: C_4$$[2, 3, 7/2, 4, 17/4]$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$