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Properties

Label 16.12.7693374828...0000.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{26}\cdot 5^{16}\cdot 7^{12}\cdot 13^{24}$
Root discriminant $3110.87$
Ramified primes $2, 5, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_{16}$ (as 16T1953)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65, 5572, -5720, -139300, 60060, 724360, -168168, -1138280, 150150, 569140, -40040, -72436, 1820, 0, 40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 40*x^14 + 1820*x^12 - 72436*x^11 - 40040*x^10 + 569140*x^9 + 150150*x^8 - 1138280*x^7 - 168168*x^6 + 724360*x^5 + 60060*x^4 - 139300*x^3 - 5720*x^2 + 5572*x + 65)
 
gp: K = bnfinit(x^16 + 40*x^14 + 1820*x^12 - 72436*x^11 - 40040*x^10 + 569140*x^9 + 150150*x^8 - 1138280*x^7 - 168168*x^6 + 724360*x^5 + 60060*x^4 - 139300*x^3 - 5720*x^2 + 5572*x + 65, 1)
 

Normalized defining polynomial

\( x^{16} + 40 x^{14} + 1820 x^{12} - 72436 x^{11} - 40040 x^{10} + 569140 x^{9} + 150150 x^{8} - 1138280 x^{7} - 168168 x^{6} + 724360 x^{5} + 60060 x^{4} - 139300 x^{3} - 5720 x^{2} + 5572 x + 65 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(76933748282119317062218648643177211504640000000000000000=2^{26}\cdot 5^{16}\cdot 7^{12}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3110.87$
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:  $2, 5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{4} - \frac{1}{10} a^{3} - \frac{3}{10} a^{2} - \frac{3}{10} a$, $\frac{1}{10} a^{5} + \frac{1}{10} a^{3} - \frac{1}{10} a^{2} + \frac{1}{5} a - \frac{1}{2}$, $\frac{1}{20} a^{6} - \frac{1}{20} a^{4} - \frac{1}{5} a^{3} + \frac{3}{20} a^{2} - \frac{1}{5} a + \frac{1}{4}$, $\frac{1}{100} a^{7} - \frac{1}{50} a^{6} - \frac{1}{20} a^{5} + \frac{1}{20} a^{3} + \frac{7}{25} a^{2} + \frac{39}{100} a + \frac{3}{10}$, $\frac{1}{200} a^{8} - \frac{1}{50} a^{6} - \frac{1}{20} a^{4} - \frac{3}{50} a^{3} - \frac{1}{10} a^{2} - \frac{3}{50} a - \frac{3}{40}$, $\frac{1}{400} a^{9} - \frac{1}{400} a^{8} - \frac{1}{200} a^{7} - \frac{1}{40} a^{6} + \frac{1}{50} a^{4} + \frac{31}{200} a^{3} + \frac{7}{200} a^{2} + \frac{31}{80} a + \frac{5}{16}$, $\frac{1}{2000} a^{10} + \frac{1}{1000} a^{9} - \frac{1}{2000} a^{8} + \frac{1}{250} a^{7} - \frac{17}{1000} a^{6} + \frac{11}{250} a^{5} - \frac{7}{1000} a^{4} + \frac{29}{250} a^{3} + \frac{369}{2000} a^{2} - \frac{181}{1000} a - \frac{93}{400}$, $\frac{1}{4000} a^{11} - \frac{1}{4000} a^{10} + \frac{3}{4000} a^{9} + \frac{1}{4000} a^{8} - \frac{9}{2000} a^{7} + \frac{7}{400} a^{6} - \frac{89}{2000} a^{5} - \frac{73}{2000} a^{4} - \frac{7}{4000} a^{3} + \frac{451}{4000} a^{2} + \frac{1111}{4000} a + \frac{389}{800}$, $\frac{1}{40000} a^{12} - \frac{1}{20000} a^{11} - \frac{1}{800} a^{9} - \frac{19}{8000} a^{8} + \frac{17}{5000} a^{7} - \frac{13}{625} a^{6} + \frac{4}{125} a^{5} - \frac{53}{1600} a^{4} - \frac{551}{4000} a^{3} + \frac{457}{1250} a^{2} + \frac{1561}{20000} a + \frac{943}{8000}$, $\frac{1}{80000} a^{13} - \frac{1}{80000} a^{12} - \frac{1}{40000} a^{11} + \frac{1}{8000} a^{10} - \frac{1}{3200} a^{9} - \frac{19}{80000} a^{8} + \frac{23}{10000} a^{7} + \frac{201}{10000} a^{6} + \frac{647}{16000} a^{5} - \frac{147}{3200} a^{4} + \frac{517}{40000} a^{3} - \frac{457}{40000} a^{2} + \frac{1717}{80000} a - \frac{6237}{16000}$, $\frac{1}{160000} a^{14} - \frac{1}{160000} a^{12} - \frac{1}{10000} a^{11} + \frac{1}{32000} a^{10} - \frac{1}{40000} a^{9} - \frac{49}{32000} a^{8} + \frac{53}{20000} a^{7} - \frac{1021}{160000} a^{6} - \frac{29}{2000} a^{5} - \frac{4771}{160000} a^{4} - \frac{959}{4000} a^{3} + \frac{47031}{160000} a^{2} + \frac{12789}{40000} a - \frac{12731}{32000}$, $\frac{1}{145600000} a^{15} + \frac{187}{145600000} a^{14} - \frac{37}{11200000} a^{13} + \frac{81}{11200000} a^{12} - \frac{1303}{11200000} a^{11} - \frac{1713}{11200000} a^{10} - \frac{12261}{11200000} a^{9} - \frac{13507}{11200000} a^{8} + \frac{29191}{11200000} a^{7} - \frac{267483}{11200000} a^{6} + \frac{161313}{11200000} a^{5} + \frac{304651}{11200000} a^{4} + \frac{1142987}{11200000} a^{3} - \frac{43011103}{145600000} a^{2} + \frac{14901589}{145600000} a + \frac{841467}{2240000}$

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:  $13$
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:  \( 1784634715930000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_{16}$ (as 16T1953):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 10461394944000
The 123 conjugacy class representatives for $A_{16}$ are not computed
Character table for $A_{16}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $15{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.7.0.1}{7} }$ R ${\href{/LocalNumberField/17.13.0.1}{13} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.11.0.1}{11} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.6.6.7$x^{6} + 2 x^{2} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
2.8.20.90$x^{8} + 16 x^{2} + 36$$8$$1$$20$$C_2^2 \wr C_2$$[2, 2, 3, 7/2]^{2}$
5Data not computed
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.6.5.1$x^{6} - 28$$6$$1$$5$$C_6$$[\ ]_{6}$
7.8.7.1$x^{8} + 14$$8$$1$$7$$D_{8}$$[\ ]_{8}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.13.24.16$x^{13} + 130 x^{12} + 13$$13$$1$$24$$C_{13}:C_3$$[2]^{3}$