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Properties

Label 16.0.30590611221...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 11^{6}\cdot 29^{4}$
Root discriminant $19.07$
Ramified primes $5, 11, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T542)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -128, 448, -64, 592, -400, 572, -138, 105, 52, -18, 50, -13, 1, 8, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 8*x^14 + x^13 - 13*x^12 + 50*x^11 - 18*x^10 + 52*x^9 + 105*x^8 - 138*x^7 + 572*x^6 - 400*x^5 + 592*x^4 - 64*x^3 + 448*x^2 - 128*x + 256)
 
gp: K = bnfinit(x^16 - 3*x^15 + 8*x^14 + x^13 - 13*x^12 + 50*x^11 - 18*x^10 + 52*x^9 + 105*x^8 - 138*x^7 + 572*x^6 - 400*x^5 + 592*x^4 - 64*x^3 + 448*x^2 - 128*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 8 x^{14} + x^{13} - 13 x^{12} + 50 x^{11} - 18 x^{10} + 52 x^{9} + 105 x^{8} - 138 x^{7} + 572 x^{6} - 400 x^{5} + 592 x^{4} - 64 x^{3} + 448 x^{2} - 128 x + 256 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(305906112217041015625=5^{12}\cdot 11^{6}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.07$
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:  $5, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{3}{8} a^{8} - \frac{3}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{48} a^{12} + \frac{1}{48} a^{11} - \frac{1}{16} a^{9} + \frac{5}{16} a^{8} - \frac{7}{24} a^{7} + \frac{5}{24} a^{6} - \frac{1}{12} a^{5} + \frac{1}{48} a^{4} - \frac{11}{24} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{96} a^{13} - \frac{1}{96} a^{12} - \frac{1}{48} a^{11} - \frac{1}{32} a^{10} + \frac{7}{32} a^{9} - \frac{11}{24} a^{8} + \frac{19}{48} a^{7} - \frac{1}{4} a^{6} - \frac{13}{32} a^{5} + \frac{1}{4} a^{4} + \frac{3}{8} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3648} a^{14} + \frac{7}{3648} a^{13} + \frac{5}{608} a^{12} + \frac{39}{1216} a^{11} - \frac{145}{1216} a^{10} + \frac{181}{912} a^{9} - \frac{769}{1824} a^{8} + \frac{67}{456} a^{7} - \frac{359}{3648} a^{6} - \frac{28}{57} a^{5} - \frac{29}{912} a^{4} - \frac{9}{19} a^{3} - \frac{11}{228} a^{2} - \frac{5}{19} a + \frac{9}{19}$, $\frac{1}{3077219991936} a^{15} - \frac{71310717}{1025739997312} a^{14} + \frac{1191123313}{769304997984} a^{13} + \frac{22054896841}{3077219991936} a^{12} - \frac{15935642089}{3077219991936} a^{11} - \frac{151735044301}{1538609995968} a^{10} - \frac{112236108867}{512869998656} a^{9} - \frac{118948434621}{256434999328} a^{8} + \frac{198962313427}{1025739997312} a^{7} + \frac{485189296793}{1538609995968} a^{6} + \frac{34376822633}{769304997984} a^{5} - \frac{49101315023}{384652498992} a^{4} + \frac{92669555263}{192326249496} a^{3} + \frac{14006050387}{48081562374} a^{2} + \frac{12174596225}{48081562374} a - \frac{5943577949}{24040781187}$

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:  $7$
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:  \( 1541.5913041 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T542):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.39875.2, 4.4.725.1, 4.2.1375.1, 8.2.5781875.1, 8.2.17490171875.1, 8.4.1590015625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
5Data 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$$[\ ]_{2}^{2}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$