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Properties

Label 16.0.89060441849...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{52}\cdot 3^{4}\cdot 5^{12}$
Root discriminant $41.87$
Ramified primes $2, 3, 5$
Class number $20$ (GRH)
Class group $[2, 10]$ (GRH)
Galois group $C_4^2:C_2$ (as 16T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![961, -1984, 128, 800, 4738, 2544, 616, 1424, 1257, 168, 592, 56, 154, -8, 24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 24*x^14 - 8*x^13 + 154*x^12 + 56*x^11 + 592*x^10 + 168*x^9 + 1257*x^8 + 1424*x^7 + 616*x^6 + 2544*x^5 + 4738*x^4 + 800*x^3 + 128*x^2 - 1984*x + 961)
 
gp: K = bnfinit(x^16 + 24*x^14 - 8*x^13 + 154*x^12 + 56*x^11 + 592*x^10 + 168*x^9 + 1257*x^8 + 1424*x^7 + 616*x^6 + 2544*x^5 + 4738*x^4 + 800*x^3 + 128*x^2 - 1984*x + 961, 1)
 

Normalized defining polynomial

\( x^{16} + 24 x^{14} - 8 x^{13} + 154 x^{12} + 56 x^{11} + 592 x^{10} + 168 x^{9} + 1257 x^{8} + 1424 x^{7} + 616 x^{6} + 2544 x^{5} + 4738 x^{4} + 800 x^{3} + 128 x^{2} - 1984 x + 961 \)

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:  \(89060441849856000000000000=2^{52}\cdot 3^{4}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.87$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31} a^{14} + \frac{14}{31} a^{13} + \frac{3}{31} a^{12} + \frac{3}{31} a^{11} + \frac{10}{31} a^{10} + \frac{10}{31} a^{9} - \frac{12}{31} a^{8} - \frac{14}{31} a^{6} - \frac{12}{31} a^{5} + \frac{14}{31} a^{4} + \frac{12}{31} a^{3} + \frac{8}{31} a^{2} + \frac{13}{31} a$, $\frac{1}{1644171511604316152403327716561} a^{15} + \frac{15800891365819971819883116597}{1644171511604316152403327716561} a^{14} - \frac{675645329591157883418084827544}{1644171511604316152403327716561} a^{13} + \frac{299355015177678067569123539352}{1644171511604316152403327716561} a^{12} + \frac{439081261889283581034384652027}{1644171511604316152403327716561} a^{11} + \frac{615229778919412388242179407032}{1644171511604316152403327716561} a^{10} + \frac{600703737687513143027699423923}{1644171511604316152403327716561} a^{9} - \frac{352541078153959870647627259656}{1644171511604316152403327716561} a^{8} - \frac{788579823156503138744065960825}{1644171511604316152403327716561} a^{7} - \frac{730615829391028861289740542702}{1644171511604316152403327716561} a^{6} - \frac{21073701499888138268233338771}{1644171511604316152403327716561} a^{5} - \frac{414920239213989685872454587149}{1644171511604316152403327716561} a^{4} + \frac{4750913994967892331316819964}{53037790696913424271075087631} a^{3} - \frac{143372300416980059573323263359}{1644171511604316152403327716561} a^{2} + \frac{75154410025975742996315589581}{1644171511604316152403327716561} a + \frac{13171639441081985375895685695}{53037790696913424271075087631}$

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

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (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:  \( 113788.60182 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4):C_4$ (as 16T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_4^2:C_2$
Character table for $C_4^2:C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.256000.4, \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.256000.2, 8.0.65536000000.1, 8.0.5898240000.1, 8.8.589824000000.1

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
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}$
3.4.0.1$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}$
5Data not computed