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Properties

Label 16.4.54566491401...0000.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{24}\cdot 5^{8}\cdot 13^{6}\cdot 29^{7}$
Root discriminant $72.20$
Ramified primes $2, 5, 13, 29$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T1558

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5372621, 17185382, -20821468, 15503434, -9192746, 4655210, -2032096, 739742, -227924, 64178, -15204, 2942, -422, 62, 0, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 62*x^13 - 422*x^12 + 2942*x^11 - 15204*x^10 + 64178*x^9 - 227924*x^8 + 739742*x^7 - 2032096*x^6 + 4655210*x^5 - 9192746*x^4 + 15503434*x^3 - 20821468*x^2 + 17185382*x - 5372621)
 
gp: K = bnfinit(x^16 - 6*x^15 + 62*x^13 - 422*x^12 + 2942*x^11 - 15204*x^10 + 64178*x^9 - 227924*x^8 + 739742*x^7 - 2032096*x^6 + 4655210*x^5 - 9192746*x^4 + 15503434*x^3 - 20821468*x^2 + 17185382*x - 5372621, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 62 x^{13} - 422 x^{12} + 2942 x^{11} - 15204 x^{10} + 64178 x^{9} - 227924 x^{8} + 739742 x^{7} - 2032096 x^{6} + 4655210 x^{5} - 9192746 x^{4} + 15503434 x^{3} - 20821468 x^{2} + 17185382 x - 5372621 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(545664914012032080281600000000=2^{24}\cdot 5^{8}\cdot 13^{6}\cdot 29^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.20$
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:  $2, 5, 13, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} + \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a$, $\frac{1}{248} a^{14} + \frac{1}{124} a^{13} - \frac{9}{248} a^{12} + \frac{1}{124} a^{11} + \frac{5}{248} a^{10} + \frac{9}{124} a^{9} + \frac{5}{248} a^{8} + \frac{4}{31} a^{7} - \frac{5}{248} a^{6} - \frac{7}{124} a^{5} + \frac{29}{248} a^{4} - \frac{49}{124} a^{3} - \frac{89}{248} a^{2} + \frac{13}{124} a + \frac{7}{248}$, $\frac{1}{659612592739699601329801199987722393718632} a^{15} - \frac{4704279752174210541677672674458257139}{164903148184924900332450299996930598429658} a^{14} - \frac{34569875428382791566923207049634707785451}{659612592739699601329801199987722393718632} a^{13} - \frac{15557649433632790580743875780031809796801}{329806296369849800664900599993861196859316} a^{12} - \frac{26592029395946584251020784195042960574117}{659612592739699601329801199987722393718632} a^{11} - \frac{37397952389801737890658511896684416280}{1070799663538473378782144805174874015777} a^{10} - \frac{9002263309720094734764306565368400463041}{94230370391385657332828742855388913388376} a^{9} - \frac{3037714286702782025520089446764957031615}{329806296369849800664900599993861196859316} a^{8} + \frac{1996952526306528485394976552174342626493}{94230370391385657332828742855388913388376} a^{7} + \frac{21758751265924394651614081361383044222043}{164903148184924900332450299996930598429658} a^{6} + \frac{25083569530052398796140861294057624737471}{659612592739699601329801199987722393718632} a^{5} + \frac{279654920293784455823502769512366054325}{2372707168128415832121587050315548178844} a^{4} + \frac{94867089177513126160865650455079825262417}{659612592739699601329801199987722393718632} a^{3} + \frac{3359202866770948761851451170733457948908}{7495597644769313651475013636224118110439} a^{2} - \frac{100363336752233260463053018505301018869773}{659612592739699601329801199987722393718632} a + \frac{148358249276518838170716699305519654148897}{329806296369849800664900599993861196859316}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $9$
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:  \( 85844127.0895 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1558:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 94 conjugacy class representatives for t16n1558 are not computed
Character table for t16n1558 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.4.659478560000.2

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 $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R $16$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
13.8.6.3$x^{8} + 65 x^{4} + 1352$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
$29$29.8.7.3$x^{8} + 58$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$