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Properties

Label 16.0.28237299218...1217.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{9}\cdot 47^{8}$
Root discriminant $33.74$
Ramified primes $17, 47$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![213, -1070, 2377, -3248, 3581, -3154, 2997, -2845, 2532, -1761, 919, -425, 175, -64, 20, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 20*x^14 - 64*x^13 + 175*x^12 - 425*x^11 + 919*x^10 - 1761*x^9 + 2532*x^8 - 2845*x^7 + 2997*x^6 - 3154*x^5 + 3581*x^4 - 3248*x^3 + 2377*x^2 - 1070*x + 213)
 
gp: K = bnfinit(x^16 - 5*x^15 + 20*x^14 - 64*x^13 + 175*x^12 - 425*x^11 + 919*x^10 - 1761*x^9 + 2532*x^8 - 2845*x^7 + 2997*x^6 - 3154*x^5 + 3581*x^4 - 3248*x^3 + 2377*x^2 - 1070*x + 213, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 20 x^{14} - 64 x^{13} + 175 x^{12} - 425 x^{11} + 919 x^{10} - 1761 x^{9} + 2532 x^{8} - 2845 x^{7} + 2997 x^{6} - 3154 x^{5} + 3581 x^{4} - 3248 x^{3} + 2377 x^{2} - 1070 x + 213 \)

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:  \(2823729921879576880531217=17^{9}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.74$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{3} a^{10} - \frac{1}{9} a^{9} - \frac{1}{3} a^{8} - \frac{2}{9} a^{7} - \frac{4}{9} a^{6} + \frac{1}{3} a^{5} + \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{2}{9} a + \frac{1}{3}$, $\frac{1}{81} a^{14} + \frac{7}{81} a^{12} - \frac{29}{81} a^{11} + \frac{20}{81} a^{10} - \frac{29}{81} a^{9} + \frac{28}{81} a^{8} - \frac{29}{81} a^{7} - \frac{2}{81} a^{6} + \frac{11}{27} a^{5} + \frac{29}{81} a^{4} - \frac{4}{9} a^{3} + \frac{1}{3} a^{2} + \frac{28}{81} a - \frac{7}{27}$, $\frac{1}{59629655841904258353} a^{15} + \frac{37447058595911042}{59629655841904258353} a^{14} + \frac{1712779793307188116}{59629655841904258353} a^{13} + \frac{2940590058720511894}{19876551947301419451} a^{12} - \frac{7870749726909612800}{59629655841904258353} a^{11} + \frac{25217963061614797160}{59629655841904258353} a^{10} + \frac{849276166402859465}{19876551947301419451} a^{9} + \frac{2429345837704842982}{6625517315767139817} a^{8} + \frac{1803637568672231869}{19876551947301419451} a^{7} + \frac{15243295878148874150}{59629655841904258353} a^{6} + \frac{23672712636264544079}{59629655841904258353} a^{5} + \frac{21929648164435001650}{59629655841904258353} a^{4} - \frac{295537252312878551}{6625517315767139817} a^{3} - \frac{23263318126081028267}{59629655841904258353} a^{2} - \frac{23353201714045053808}{59629655841904258353} a - \frac{17826536715132847}{279951435877484781}$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  \( 181509.481996 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-47}) \), 4.0.37553.1, 8.0.23973872753.2

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
$47$47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$