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Properties

Label 16.16.3143861048...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 13^{4}$
Root discriminant $29.42$
Ramified primes $2, 3, 5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2 \times (C_4\times C_2):C_2$ (as 16T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-71, 104, 1424, 276, -5357, -1052, 8646, -208, -6664, 1720, 2032, -828, -226, 140, 0, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 140*x^13 - 226*x^12 - 828*x^11 + 2032*x^10 + 1720*x^9 - 6664*x^8 - 208*x^7 + 8646*x^6 - 1052*x^5 - 5357*x^4 + 276*x^3 + 1424*x^2 + 104*x - 71)
 
gp: K = bnfinit(x^16 - 8*x^15 + 140*x^13 - 226*x^12 - 828*x^11 + 2032*x^10 + 1720*x^9 - 6664*x^8 - 208*x^7 + 8646*x^6 - 1052*x^5 - 5357*x^4 + 276*x^3 + 1424*x^2 + 104*x - 71, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 140 x^{13} - 226 x^{12} - 828 x^{11} + 2032 x^{10} + 1720 x^{9} - 6664 x^{8} - 208 x^{7} + 8646 x^{6} - 1052 x^{5} - 5357 x^{4} + 276 x^{3} + 1424 x^{2} + 104 x - 71 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(314386104818073600000000=2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 13^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.42$
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:  $2, 3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1633} a^{14} - \frac{7}{1633} a^{13} + \frac{103}{1633} a^{12} - \frac{527}{1633} a^{11} + \frac{779}{1633} a^{10} + \frac{769}{1633} a^{9} + \frac{309}{1633} a^{8} + \frac{70}{1633} a^{7} - \frac{365}{1633} a^{6} + \frac{595}{1633} a^{5} + \frac{118}{1633} a^{4} - \frac{804}{1633} a^{3} + \frac{287}{1633} a^{2} + \frac{305}{1633} a + \frac{9}{23}$, $\frac{1}{4505447} a^{15} + \frac{1372}{4505447} a^{14} + \frac{1336042}{4505447} a^{13} + \frac{406056}{4505447} a^{12} - \frac{53444}{145337} a^{11} - \frac{1220988}{4505447} a^{10} - \frac{67480}{195889} a^{9} + \frac{24935}{50623} a^{8} - \frac{2198200}{4505447} a^{7} + \frac{186386}{4505447} a^{6} + \frac{1086802}{4505447} a^{5} + \frac{1270725}{4505447} a^{4} - \frac{1711006}{4505447} a^{3} + \frac{1710643}{4505447} a^{2} + \frac{2080362}{4505447} a + \frac{30351}{63457}$

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:  $15$
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:  \( 2544047.1534 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4:C_2$ (as 16T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2 \times (C_4\times C_2):C_2$
Character table for $C_2 \times (C_4\times C_2):C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{3}, \sqrt{5})\), 8.8.3317760000.1, 8.8.560701440000.1, 8.8.432640000.1

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

Sibling fields

Galois closure: data not computed
Degree 16 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$