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Properties

Label 16.0.94276032870...4624.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{42}\cdot 11^{8}$
Root discriminant $20.46$
Ramified primes $2, 11$
Class number $1$
Class group Trivial
Galois group $C_2^3:S_4.C_2$ (as 16T764)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 24, 152, 432, 408, -224, -340, 96, 94, -72, 16, 44, -12, -8, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^13 - 12*x^12 + 44*x^11 + 16*x^10 - 72*x^9 + 94*x^8 + 96*x^7 - 340*x^6 - 224*x^5 + 408*x^4 + 432*x^3 + 152*x^2 + 24*x + 2)
 
gp: K = bnfinit(x^16 - 8*x^13 - 12*x^12 + 44*x^11 + 16*x^10 - 72*x^9 + 94*x^8 + 96*x^7 - 340*x^6 - 224*x^5 + 408*x^4 + 432*x^3 + 152*x^2 + 24*x + 2, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{13} - 12 x^{12} + 44 x^{11} + 16 x^{10} - 72 x^{9} + 94 x^{8} + 96 x^{7} - 340 x^{6} - 224 x^{5} + 408 x^{4} + 432 x^{3} + 152 x^{2} + 24 x + 2 \)

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:  \(942760328706207514624=2^{42}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.46$
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:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{29} a^{14} + \frac{2}{29} a^{13} + \frac{9}{29} a^{12} - \frac{9}{29} a^{11} - \frac{14}{29} a^{10} + \frac{4}{29} a^{8} - \frac{6}{29} a^{7} - \frac{14}{29} a^{6} + \frac{9}{29} a^{5} + \frac{14}{29} a^{4} - \frac{6}{29} a^{3} + \frac{2}{29} a^{2} - \frac{12}{29}$, $\frac{1}{188975800882217} a^{15} - \frac{1909159598072}{188975800882217} a^{14} + \frac{13893603221737}{188975800882217} a^{13} - \frac{88411892480926}{188975800882217} a^{12} + \frac{2667622393475}{188975800882217} a^{11} - \frac{207134302230}{979149227369} a^{10} + \frac{5444319790978}{188975800882217} a^{9} - \frac{7953190565522}{188975800882217} a^{8} - \frac{89178111177331}{188975800882217} a^{7} + \frac{86571268614768}{188975800882217} a^{6} + \frac{55099522117690}{188975800882217} a^{5} - \frac{82386053334}{188975800882217} a^{4} + \frac{8722562465236}{188975800882217} a^{3} + \frac{66609924107590}{188975800882217} a^{2} - \frac{79310859712062}{188975800882217} a - \frac{72939649920694}{188975800882217}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{421989280}{1164267809} a^{15} - \frac{133572846}{1164267809} a^{14} + \frac{66159228}{1164267809} a^{13} - \frac{3419813912}{1164267809} a^{12} - \frac{3960399550}{1164267809} a^{11} + \frac{19584502620}{1164267809} a^{10} + \frac{463848000}{1164267809} a^{9} - \frac{29482063327}{1164267809} a^{8} + \frac{48476184296}{1164267809} a^{7} + \frac{24203509050}{1164267809} a^{6} - \frac{148198541064}{1164267809} a^{5} - \frac{48125025428}{1164267809} a^{4} + \frac{180021129892}{1164267809} a^{3} + \frac{125178016100}{1164267809} a^{2} + \frac{31722779148}{1164267809} a + \frac{3403576863}{1164267809} \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 17322.7614792 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:S_4.C_2$ (as 16T764):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 23 conjugacy class representatives for $C_2^3:S_4.C_2$
Character table for $C_2^3:S_4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.2.2816.1, 8.0.31719424.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
2Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$