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Properties

Label 16.0.52205595918...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 3^{4}\cdot 5^{8}\cdot 7^{4}$
Root discriminant $22.77$
Ramified primes $2, 3, 5, 7$
Class number $4$
Class group $[2, 2]$
Galois group $C_2\times Q_8:C_2^2$ (as 16T69)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![961, -3224, 5472, -6760, 7104, -6384, 4898, -2944, 1343, -536, 170, 12, -30, 4, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 + 4*x^13 - 30*x^12 + 12*x^11 + 170*x^10 - 536*x^9 + 1343*x^8 - 2944*x^7 + 4898*x^6 - 6384*x^5 + 7104*x^4 - 6760*x^3 + 5472*x^2 - 3224*x + 961)
 
gp: K = bnfinit(x^16 - 4*x^15 + 6*x^14 + 4*x^13 - 30*x^12 + 12*x^11 + 170*x^10 - 536*x^9 + 1343*x^8 - 2944*x^7 + 4898*x^6 - 6384*x^5 + 7104*x^4 - 6760*x^3 + 5472*x^2 - 3224*x + 961, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 6 x^{14} + 4 x^{13} - 30 x^{12} + 12 x^{11} + 170 x^{10} - 536 x^{9} + 1343 x^{8} - 2944 x^{7} + 4898 x^{6} - 6384 x^{5} + 7104 x^{4} - 6760 x^{3} + 5472 x^{2} - 3224 x + 961 \)

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:  \(5220559591833600000000=2^{36}\cdot 3^{4}\cdot 5^{8}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.77$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1085} a^{14} + \frac{96}{217} a^{13} + \frac{477}{1085} a^{12} - \frac{388}{1085} a^{11} - \frac{6}{31} a^{10} + \frac{424}{1085} a^{9} + \frac{321}{1085} a^{8} - \frac{48}{1085} a^{7} - \frac{313}{1085} a^{6} - \frac{459}{1085} a^{5} + \frac{424}{1085} a^{4} - \frac{2}{1085} a^{3} - \frac{19}{217} a^{2} - \frac{257}{1085} a - \frac{16}{35}$, $\frac{1}{39102249320118455218475} a^{15} + \frac{16203918486260167498}{39102249320118455218475} a^{14} + \frac{5052508901869289604132}{39102249320118455218475} a^{13} + \frac{7131275781455470994618}{39102249320118455218475} a^{12} + \frac{2947343318291816948416}{39102249320118455218475} a^{11} + \frac{11303937954393504795554}{39102249320118455218475} a^{10} - \frac{372830779316532322497}{39102249320118455218475} a^{9} + \frac{3022150119863372423743}{7820449864023691043695} a^{8} - \frac{13767226785926685201647}{39102249320118455218475} a^{7} + \frac{11209762722689932886997}{39102249320118455218475} a^{6} + \frac{13359761299446014175227}{39102249320118455218475} a^{5} - \frac{238010435237529989550}{1564089972804738208739} a^{4} + \frac{32626075885416708361}{69950356565507075525} a^{3} + \frac{504097878090695231281}{3007865332316804247575} a^{2} - \frac{10631415673955912001622}{39102249320118455218475} a - \frac{239389528305799873013}{1261362881294143716725}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( \frac{110760213526901669858}{39102249320118455218475} a^{15} - \frac{408153835325711414536}{39102249320118455218475} a^{14} + \frac{553908133380467601531}{39102249320118455218475} a^{13} + \frac{611130072714456398479}{39102249320118455218475} a^{12} - \frac{3192771348254505902062}{39102249320118455218475} a^{11} + \frac{521647822046338798157}{39102249320118455218475} a^{10} + \frac{18969674933105708015494}{39102249320118455218475} a^{9} - \frac{311361764131532137417}{223441424686391172677} a^{8} + \frac{133363816425398432818234}{39102249320118455218475} a^{7} - \frac{283455541207589392924739}{39102249320118455218475} a^{6} + \frac{65226321344293867214053}{5586035617159779316925} a^{5} - \frac{112387540241554003071836}{7820449864023691043695} a^{4} + \frac{1059778024813514658973}{69950356565507075525} a^{3} - \frac{40720786130774083416402}{3007865332316804247575} a^{2} + \frac{393683261947098703413564}{39102249320118455218475} a - \frac{5798675197643267780584}{1261362881294143716725} \) (order $8$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 18484.9595466 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times Q_8:C_2^2$ (as 16T69):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 34 conjugacy class representatives for $C_2\times Q_8:C_2^2$
Character table for $C_2\times Q_8:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), 8.0.40960000.1, 8.0.18063360000.1, 8.8.18063360000.1

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 R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$