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Properties

Label 16.0.14390210149...8481.3
Degree $16$
Signature $[0, 8]$
Discriminant $17^{12}\cdot 89^{12}$
Root discriminant $242.59$
Ramified primes $17, 89$
Class number $20384$ (GRH)
Class group $[28, 728]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2548078528, 646882304, 30548912, -404797504, 85842500, 13350656, -2897647, -1172884, 478829, 37482, -33967, -2202, 1509, 154, -39, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 39*x^14 + 154*x^13 + 1509*x^12 - 2202*x^11 - 33967*x^10 + 37482*x^9 + 478829*x^8 - 1172884*x^7 - 2897647*x^6 + 13350656*x^5 + 85842500*x^4 - 404797504*x^3 + 30548912*x^2 + 646882304*x + 2548078528)
 
gp: K = bnfinit(x^16 - 4*x^15 - 39*x^14 + 154*x^13 + 1509*x^12 - 2202*x^11 - 33967*x^10 + 37482*x^9 + 478829*x^8 - 1172884*x^7 - 2897647*x^6 + 13350656*x^5 + 85842500*x^4 - 404797504*x^3 + 30548912*x^2 + 646882304*x + 2548078528, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 39 x^{14} + 154 x^{13} + 1509 x^{12} - 2202 x^{11} - 33967 x^{10} + 37482 x^{9} + 478829 x^{8} - 1172884 x^{7} - 2897647 x^{6} + 13350656 x^{5} + 85842500 x^{4} - 404797504 x^{3} + 30548912 x^{2} + 646882304 x + 2548078528 \)

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:  \(143902101499474565788603458614754288481=17^{12}\cdot 89^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $242.59$
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:  $17, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{3}$, $\frac{1}{32} a^{10} + \frac{1}{32} a^{9} + \frac{1}{32} a^{8} + \frac{1}{16} a^{7} + \frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{3}{32} a^{4} - \frac{3}{32} a^{3} + \frac{13}{32} a^{2} + \frac{3}{8} a + \frac{3}{8}$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} + \frac{3}{64} a^{9} - \frac{1}{16} a^{8} + \frac{3}{32} a^{7} + \frac{1}{32} a^{6} - \frac{7}{64} a^{5} + \frac{11}{64} a^{4} - \frac{25}{64} a^{3} - \frac{9}{32} a^{2} + \frac{5}{16} a + \frac{3}{8}$, $\frac{1}{5092672} a^{12} - \frac{3}{5092672} a^{11} + \frac{9325}{5092672} a^{10} + \frac{79583}{2546336} a^{9} - \frac{114217}{2546336} a^{8} + \frac{307771}{2546336} a^{7} + \frac{555973}{5092672} a^{6} + \frac{975881}{5092672} a^{5} + \frac{731193}{5092672} a^{4} - \frac{287087}{636584} a^{3} - \frac{145143}{636584} a^{2} - \frac{11312}{79573} a + \frac{137107}{318292}$, $\frac{1}{10185344} a^{13} - \frac{70257}{10185344} a^{11} - \frac{25789}{5092672} a^{10} + \frac{328637}{10185344} a^{9} + \frac{70853}{1273168} a^{8} - \frac{1257759}{10185344} a^{7} - \frac{349133}{5092672} a^{6} + \frac{2306095}{10185344} a^{5} + \frac{315349}{1273168} a^{4} + \frac{4441729}{10185344} a^{3} - \frac{2501565}{5092672} a^{2} - \frac{74121}{2546336} a + \frac{424777}{1273168}$, $\frac{1}{81482752} a^{14} - \frac{1}{20370688} a^{13} - \frac{3}{81482752} a^{12} + \frac{88917}{40741376} a^{11} + \frac{449417}{81482752} a^{10} + \frac{726543}{40741376} a^{9} + \frac{174853}{81482752} a^{8} + \frac{3212097}{40741376} a^{7} + \frac{7769857}{81482752} a^{6} + \frac{1409053}{20370688} a^{5} - \frac{1566375}{6267904} a^{4} - \frac{447945}{2546336} a^{3} + \frac{4658525}{10185344} a^{2} - \frac{413561}{1273168} a - \frac{2517059}{5092672}$, $\frac{1}{196793114362898970812296281557807104} a^{15} + \frac{143693732442656209019305077}{98396557181449485406148140778903552} a^{14} + \frac{4423463314997383587890105109}{196793114362898970812296281557807104} a^{13} + \frac{475683994153394203149895703}{6149784823840592837884258798681472} a^{12} + \frac{351500286020459090303887938013909}{196793114362898970812296281557807104} a^{11} + \frac{447861351197524945070027565372099}{49198278590724742703074070389451776} a^{10} - \frac{151291014571432204172890858489779}{15137931874069151600945867812139008} a^{9} + \frac{1264908696314576766032234582677189}{24599139295362371351537035194725888} a^{8} - \frac{19905823718119691851026981870312307}{196793114362898970812296281557807104} a^{7} - \frac{3913443041016181659153812048524815}{98396557181449485406148140778903552} a^{6} + \frac{28928397654212138098829383514920861}{196793114362898970812296281557807104} a^{5} - \frac{19775044974894504648065541451359941}{98396557181449485406148140778903552} a^{4} - \frac{4958862348695792387946050261905195}{24599139295362371351537035194725888} a^{3} - \frac{294530504627317118790413664268385}{12299569647681185675768517597362944} a^{2} - \frac{3815128511137829731636940824533275}{12299569647681185675768517597362944} a + \frac{664581738279290098091949086201299}{6149784823840592837884258798681472}$

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

Class group and class number

$C_{28}\times C_{728}$, which has order $20384$ (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:  \( 651512602.568 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{89}) \), \(\Q(\sqrt{1513}) \), \(\Q(\sqrt{17}) \), 4.4.704969.1, \(\Q(\sqrt{17}, \sqrt{89})\), 4.4.203736041.1, 4.0.38915873.1 x2, 4.0.437257.1 x2, 8.8.41508374402353681.1, 8.0.1514445171352129.3, 8.0.11995920202280213809.4

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$89$89.8.6.1$x^{8} - 4361 x^{4} + 10265616$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
89.8.6.1$x^{8} - 4361 x^{4} + 10265616$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$