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Properties

Label 16.0.18397442524...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{14}\cdot 7^{8}\cdot 41^{6}$
Root discriminant $213.33$
Ramified primes $2, 3, 5, 7, 41$
Class number $96$ (GRH)
Class group $[2, 4, 12]$ (GRH)
Galois group 16T813

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37449990400, 0, -8271372800, 0, 910331200, 0, -59208800, 0, 2828640, 0, -102040, 0, 3220, 0, -70, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 70*x^14 + 3220*x^12 - 102040*x^10 + 2828640*x^8 - 59208800*x^6 + 910331200*x^4 - 8271372800*x^2 + 37449990400)
 
gp: K = bnfinit(x^16 - 70*x^14 + 3220*x^12 - 102040*x^10 + 2828640*x^8 - 59208800*x^6 + 910331200*x^4 - 8271372800*x^2 + 37449990400, 1)
 

Normalized defining polynomial

\( x^{16} - 70 x^{14} + 3220 x^{12} - 102040 x^{10} + 2828640 x^{8} - 59208800 x^{6} + 910331200 x^{4} - 8271372800 x^{2} + 37449990400 \)

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:  \(18397442524921103360102400000000000000=2^{24}\cdot 3^{8}\cdot 5^{14}\cdot 7^{8}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $213.33$
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:  $2, 3, 5, 7, 41$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{80} a^{8}$, $\frac{1}{80} a^{9}$, $\frac{1}{1760} a^{10} + \frac{3}{880} a^{8} + \frac{1}{88} a^{6} - \frac{5}{44} a^{4} + \frac{2}{11} a^{2} + \frac{2}{11}$, $\frac{1}{1760} a^{11} + \frac{3}{880} a^{9} + \frac{1}{88} a^{7} - \frac{5}{44} a^{5} + \frac{2}{11} a^{3} + \frac{2}{11} a$, $\frac{1}{542080} a^{12} + \frac{3}{13552} a^{10} - \frac{9}{6160} a^{8} - \frac{355}{13552} a^{6} + \frac{775}{6776} a^{4} - \frac{485}{3388} a^{2} - \frac{799}{1694}$, $\frac{1}{542080} a^{13} + \frac{3}{13552} a^{11} - \frac{9}{6160} a^{9} - \frac{355}{13552} a^{7} + \frac{775}{6776} a^{5} - \frac{485}{3388} a^{3} - \frac{799}{1694} a$, $\frac{1}{207330324925262739200} a^{14} + \frac{356434747711}{2961861784646610560} a^{12} + \frac{215413493481013}{1295814530782892120} a^{10} + \frac{7167723659592823}{5183258123131568480} a^{8} - \frac{53824490527605243}{1295814530782892120} a^{6} + \frac{5392835686591277}{64790726539144606} a^{4} - \frac{19270802674399}{112875830207569} a^{2} - \frac{784629363536163}{1580261622905966}$, $\frac{1}{12232489170590501612800} a^{15} - \frac{30526458759617}{34949969058830004608} a^{13} + \frac{4530487961647779}{30581222926476254032} a^{11} + \frac{1261751792099392921}{305812229264762540320} a^{9} + \frac{3223194074923345077}{76453057316190635080} a^{7} - \frac{689864458184426677}{7645305731619063508} a^{5} + \frac{966760209965274}{6659673982246571} a^{3} + \frac{8592460266600065}{93235435751451994} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{12}$, which has order $96$ (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:  \( 18349122010.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T813:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n813
Character table for t16n813 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{21}) \), 4.0.46125.1, 4.0.251125.1, \(\Q(\sqrt{5}, \sqrt{21})\), 8.0.5108165015625.3

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 R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$2$2.8.12.4$x^{8} + 2 x^{6} + 16$$2$$4$$12$$C_2^3: C_4$$[2, 2, 3]^{4}$
2.8.12.4$x^{8} + 2 x^{6} + 16$$2$$4$$12$$C_2^3: C_4$$[2, 2, 3]^{4}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
41Data not computed