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Properties

Label 16.0.69012051254...3184.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 13^{12}\cdot 17^{14}$
Root discriminant $549.46$
Ramified primes $2, 13, 17$
Class number $1217545216$ (GRH)
Class group $[2, 2, 2, 2, 2, 4, 9512072]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51588306250000, 0, 20635322500000, 0, 2669893265000, 0, 134351822800, 0, 3291492672, 0, 42405480, 0, 283322, 0, 884, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 884*x^14 + 283322*x^12 + 42405480*x^10 + 3291492672*x^8 + 134351822800*x^6 + 2669893265000*x^4 + 20635322500000*x^2 + 51588306250000)
 
gp: K = bnfinit(x^16 + 884*x^14 + 283322*x^12 + 42405480*x^10 + 3291492672*x^8 + 134351822800*x^6 + 2669893265000*x^4 + 20635322500000*x^2 + 51588306250000, 1)
 

Normalized defining polynomial

\( x^{16} + 884 x^{14} + 283322 x^{12} + 42405480 x^{10} + 3291492672 x^{8} + 134351822800 x^{6} + 2669893265000 x^{4} + 20635322500000 x^{2} + 51588306250000 \)

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:  \(69012051254784469945608052647868865324253184=2^{44}\cdot 13^{12}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $549.46$
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:  $2, 13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3536=2^{4}\cdot 13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{3536}(1,·)$, $\chi_{3536}(1477,·)$, $\chi_{3536}(1665,·)$, $\chi_{3536}(2313,·)$, $\chi_{3536}(525,·)$, $\chi_{3536}(1041,·)$, $\chi_{3536}(1685,·)$, $\chi_{3536}(441,·)$, $\chi_{3536}(3353,·)$, $\chi_{3536}(733,·)$, $\chi_{3536}(229,·)$, $\chi_{3536}(1981,·)$, $\chi_{3536}(625,·)$, $\chi_{3536}(2933,·)$, $\chi_{3536}(2937,·)$, $\chi_{3536}(2813,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{26} a^{4}$, $\frac{1}{26} a^{5}$, $\frac{1}{26} a^{6}$, $\frac{1}{26} a^{7}$, $\frac{1}{11492} a^{8}$, $\frac{1}{57460} a^{9} + \frac{1}{65} a^{7} + \frac{1}{130} a^{5} + \frac{1}{5} a$, $\frac{1}{287300} a^{10} + \frac{9}{287300} a^{8} - \frac{9}{650} a^{6} - \frac{1}{65} a^{4} - \frac{9}{25} a^{2}$, $\frac{1}{1436500} a^{11} + \frac{9}{1436500} a^{9} + \frac{8}{1625} a^{7} - \frac{6}{325} a^{5} + \frac{41}{125} a^{3} + \frac{1}{5} a$, $\frac{1}{19421480000} a^{12} + \frac{12}{23343125} a^{10} + \frac{93}{7182500} a^{8} - \frac{3071}{169000} a^{6} + \frac{2}{8125} a^{4} - \frac{31}{65} a^{2} + \frac{49}{104}$, $\frac{1}{97107400000} a^{13} + \frac{12}{116715625} a^{11} + \frac{93}{35912500} a^{9} - \frac{3071}{845000} a^{7} - \frac{623}{40625} a^{5} - \frac{161}{325} a^{3} + \frac{49}{520} a$, $\frac{1}{44405534645109937000000} a^{14} + \frac{724671602809}{44405534645109937000000} a^{12} + \frac{130600854944967}{106744073666129656250} a^{10} - \frac{45935512303117}{6568866071761825000} a^{8} - \frac{31489814714035167}{1932019432871125000} a^{6} - \frac{3939222883633}{1486168794516250} a^{4} + \frac{4338628420941}{18291308240200} a^{2} - \frac{1216546062231}{9511480284904}$, $\frac{1}{222027673225549685000000} a^{15} + \frac{724671602809}{222027673225549685000000} a^{13} + \frac{130600854944967}{533720368330648281250} a^{11} - \frac{45935512303117}{32844330358809125000} a^{9} + \frac{42818625011777333}{9660097164355625000} a^{7} - \frac{118259899384883}{7430843972581250} a^{5} + \frac{40921244901341}{91456541201000} a^{3} - \frac{2145605269427}{9511480284904} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{9512072}$, which has order $1217545216$ (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:  \( 2510752.6365781664 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{442}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{17}, \sqrt{26})\), 4.4.53139008.1, 4.4.4913.1, 8.8.2823754171224064.16, 8.0.8307349231540977532928.4, 8.0.8307349231540977532928.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$13$13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$