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Properties

Label 16.8.15449861545...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{8}\cdot 13^{8}\cdot 17^{2}$
Root discriminant $32.49$
Ramified primes $2, 5, 13, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1123

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-256, 2304, 6080, -1536, -13376, -8592, 3832, 5488, 951, -582, -115, 12, -40, -18, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^14 - 18*x^13 - 40*x^12 + 12*x^11 - 115*x^10 - 582*x^9 + 951*x^8 + 5488*x^7 + 3832*x^6 - 8592*x^5 - 13376*x^4 - 1536*x^3 + 6080*x^2 + 2304*x - 256)
 
gp: K = bnfinit(x^16 - 2*x^14 - 18*x^13 - 40*x^12 + 12*x^11 - 115*x^10 - 582*x^9 + 951*x^8 + 5488*x^7 + 3832*x^6 - 8592*x^5 - 13376*x^4 - 1536*x^3 + 6080*x^2 + 2304*x - 256, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{14} - 18 x^{13} - 40 x^{12} + 12 x^{11} - 115 x^{10} - 582 x^{9} + 951 x^{8} + 5488 x^{7} + 3832 x^{6} - 8592 x^{5} - 13376 x^{4} - 1536 x^{3} + 6080 x^{2} + 2304 x - 256 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1544986154559078400000000=2^{24}\cdot 5^{8}\cdot 13^{8}\cdot 17^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.49$
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:  $2, 5, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{3}{16} a^{6} - \frac{3}{8} a^{5} + \frac{7}{16} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{544} a^{13} - \frac{7}{272} a^{12} - \frac{15}{272} a^{11} - \frac{27}{272} a^{10} + \frac{15}{136} a^{9} - \frac{27}{136} a^{8} + \frac{229}{544} a^{7} + \frac{45}{136} a^{6} - \frac{33}{544} a^{5} - \frac{69}{272} a^{4} - \frac{47}{136} a^{3} - \frac{4}{17} a^{2} + \frac{11}{34} a - \frac{5}{17}$, $\frac{1}{1088} a^{14} - \frac{11}{544} a^{12} - \frac{33}{544} a^{11} - \frac{1}{68} a^{10} + \frac{13}{272} a^{9} - \frac{467}{1088} a^{8} - \frac{75}{544} a^{7} + \frac{243}{1088} a^{6} - \frac{41}{136} a^{5} + \frac{31}{272} a^{4} + \frac{3}{34} a^{3} - \frac{33}{68} a^{2} - \frac{13}{34} a - \frac{1}{17}$, $\frac{1}{74406861560407728512} a^{15} + \frac{3703711007117871}{9300857695050966064} a^{14} + \frac{8821116768977541}{37203430780203864256} a^{13} + \frac{153635070258757459}{37203430780203864256} a^{12} + \frac{1824822398641737}{113425093842084952} a^{11} + \frac{1130372247567732793}{18601715390101932128} a^{10} - \frac{11212321716923409091}{74406861560407728512} a^{9} - \frac{61826571255264315}{1282876923455305664} a^{8} - \frac{12526908915226604445}{74406861560407728512} a^{7} + \frac{3494338174351107199}{9300857695050966064} a^{6} - \frac{5591278303777866313}{18601715390101932128} a^{5} - \frac{1240319175227292535}{9300857695050966064} a^{4} - \frac{618041460760565197}{2325214423762741516} a^{3} - \frac{113186702847580693}{1162607211881370758} a^{2} - \frac{3319968736036891}{20044951928989151} a + \frac{146990438919959093}{581303605940685379}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 614564.722407 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1123:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1123 are not computed
Character table for t16n1123 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.432640000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$