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Properties

Label 16.0.43292507018...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{2}\cdot 5^{12}\cdot 19^{6}\cdot 61^{2}\cdot 103^{4}$
Root discriminant $61.63$
Ramified primes $3, 5, 19, 61, 103$
Class number $2848$ (GRH)
Class group $[2, 2, 2, 2, 178]$ (GRH)
Galois group 16T1046

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![705681, 605979, 1103694, 496341, 728516, 169817, 235141, 29331, 54241, -834, 9390, -636, 952, -52, 50, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 50*x^14 - 52*x^13 + 952*x^12 - 636*x^11 + 9390*x^10 - 834*x^9 + 54241*x^8 + 29331*x^7 + 235141*x^6 + 169817*x^5 + 728516*x^4 + 496341*x^3 + 1103694*x^2 + 605979*x + 705681)
 
gp: K = bnfinit(x^16 - x^15 + 50*x^14 - 52*x^13 + 952*x^12 - 636*x^11 + 9390*x^10 - 834*x^9 + 54241*x^8 + 29331*x^7 + 235141*x^6 + 169817*x^5 + 728516*x^4 + 496341*x^3 + 1103694*x^2 + 605979*x + 705681, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 50 x^{14} - 52 x^{13} + 952 x^{12} - 636 x^{11} + 9390 x^{10} - 834 x^{9} + 54241 x^{8} + 29331 x^{7} + 235141 x^{6} + 169817 x^{5} + 728516 x^{4} + 496341 x^{3} + 1103694 x^{2} + 605979 x + 705681 \)

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:  \(43292507018833059260009765625=3^{2}\cdot 5^{12}\cdot 19^{6}\cdot 61^{2}\cdot 103^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.63$
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:  $3, 5, 19, 61, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{53904183783284312713636633751858421112378707} a^{15} + \frac{1597602973846449613944900880086107521186737}{17968061261094770904545544583952807037459569} a^{14} + \frac{12691818713602519551396425225905322388193585}{53904183783284312713636633751858421112378707} a^{13} + \frac{20911006407908590786573132968662254772130820}{53904183783284312713636633751858421112378707} a^{12} - \frac{4205739025133704167609250264632263078317401}{17968061261094770904545544583952807037459569} a^{11} + \frac{8162142417783613413808260112437581102882017}{53904183783284312713636633751858421112378707} a^{10} - \frac{4854307358928054838570745075944479873541145}{17968061261094770904545544583952807037459569} a^{9} + \frac{7536307868417036764558585153094142176886189}{17968061261094770904545544583952807037459569} a^{8} + \frac{252794262183286397171896470239477193398590}{53904183783284312713636633751858421112378707} a^{7} - \frac{6562742400488715057400755331181900885165483}{53904183783284312713636633751858421112378707} a^{6} - \frac{3647895059065392673763868181244974523257055}{53904183783284312713636633751858421112378707} a^{5} - \frac{7762633219283480510926794006171239170740341}{17968061261094770904545544583952807037459569} a^{4} - \frac{7891319227359299895914264476355258971007393}{53904183783284312713636633751858421112378707} a^{3} - \frac{22703743901084437495003220853412689969200203}{53904183783284312713636633751858421112378707} a^{2} - \frac{5217927461959652523076296173210625714862689}{17968061261094770904545544583952807037459569} a + \frac{3435275981142854452978123717356932347662067}{17968061261094770904545544583952807037459569}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{178}$, which has order $2848$ (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:  \( 10491.0986114 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1046:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 768
The 40 conjugacy class representatives for t16n1046
Character table for t16n1046 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.1957.1, 8.8.2393655625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
5Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$61$61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.6.0.1$x^{6} - 4 x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$
61.6.0.1$x^{6} - 4 x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$
$103$103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.8.4.1$x^{8} + 106090 x^{4} - 1092727 x^{2} + 2813772025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$