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Properties

Label 16.8.42234996556...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{11}\cdot 29^{3}\cdot 101^{7}\cdot 149^{3}$
Root discriminant $109.42$
Ramified primes $5, 29, 101, 149$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1642

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![120371, 46166, -861001, -459358, 1695667, 1051474, -516604, -112204, -6105, -66076, 35444, -8514, 3382, -332, 102, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 102*x^14 - 332*x^13 + 3382*x^12 - 8514*x^11 + 35444*x^10 - 66076*x^9 - 6105*x^8 - 112204*x^7 - 516604*x^6 + 1051474*x^5 + 1695667*x^4 - 459358*x^3 - 861001*x^2 + 46166*x + 120371)
 
gp: K = bnfinit(x^16 - 4*x^15 + 102*x^14 - 332*x^13 + 3382*x^12 - 8514*x^11 + 35444*x^10 - 66076*x^9 - 6105*x^8 - 112204*x^7 - 516604*x^6 + 1051474*x^5 + 1695667*x^4 - 459358*x^3 - 861001*x^2 + 46166*x + 120371, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 102 x^{14} - 332 x^{13} + 3382 x^{12} - 8514 x^{11} + 35444 x^{10} - 66076 x^{9} - 6105 x^{8} - 112204 x^{7} - 516604 x^{6} + 1051474 x^{5} + 1695667 x^{4} - 459358 x^{3} - 861001 x^{2} + 46166 x + 120371 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(422349965563628096879436572265625=5^{11}\cdot 29^{3}\cdot 101^{7}\cdot 149^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $109.42$
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:  $5, 29, 101, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{202} a^{12} - \frac{24}{101} a^{11} - \frac{27}{202} a^{10} - \frac{21}{202} a^{9} - \frac{3}{202} a^{8} - \frac{35}{101} a^{7} + \frac{45}{101} a^{6} - \frac{48}{101} a^{5} - \frac{39}{101} a^{4} + \frac{9}{101} a^{3} - \frac{3}{101} a^{2} + \frac{67}{202} a - \frac{10}{101}$, $\frac{1}{202} a^{13} - \frac{4}{101} a^{11} - \frac{2}{101} a^{10} - \frac{1}{202} a^{9} - \frac{6}{101} a^{8} + \frac{63}{202} a^{7} + \frac{83}{202} a^{6} + \frac{61}{202} a^{5} - \frac{45}{101} a^{4} + \frac{25}{101} a^{3} + \frac{41}{101} a^{2} - \frac{18}{101} a + \frac{25}{101}$, $\frac{1}{20402} a^{14} - \frac{13}{10201} a^{13} + \frac{24}{10201} a^{12} + \frac{4687}{20402} a^{11} + \frac{2833}{20402} a^{10} + \frac{4393}{20402} a^{9} + \frac{760}{10201} a^{8} + \frac{3373}{10201} a^{7} - \frac{4329}{20402} a^{6} + \frac{1827}{10201} a^{5} - \frac{4423}{10201} a^{4} - \frac{408}{10201} a^{3} - \frac{848}{10201} a^{2} + \frac{3581}{10201} a + \frac{2327}{20402}$, $\frac{1}{1262523554582610654755690700228675818} a^{15} - \frac{311378830098503739766783759535}{1262523554582610654755690700228675818} a^{14} + \frac{1068525389678573136774436504535462}{631261777291305327377845350114337909} a^{13} + \frac{148091851026446473649627054936045}{631261777291305327377845350114337909} a^{12} + \frac{58722137633361881648674222394777621}{631261777291305327377845350114337909} a^{11} + \frac{145817977494888229410205587838311271}{1262523554582610654755690700228675818} a^{10} - \frac{268488190685272382188372259080823037}{1262523554582610654755690700228675818} a^{9} + \frac{90212855432033804487333754076823837}{1262523554582610654755690700228675818} a^{8} - \frac{463893737003198480597925156505582137}{1262523554582610654755690700228675818} a^{7} - \frac{315922079460365498848255854358244763}{1262523554582610654755690700228675818} a^{6} + \frac{103546586409911619069900348694060711}{1262523554582610654755690700228675818} a^{5} - \frac{289455539051747628117197876275858439}{631261777291305327377845350114337909} a^{4} + \frac{335062958719334471841773584908273911}{1262523554582610654755690700228675818} a^{3} - \frac{504739658149523164375215991613033097}{1262523554582610654755690700228675818} a^{2} - \frac{111793728478748662070032806132948279}{1262523554582610654755690700228675818} a - \frac{229750742915588841927117366758163395}{1262523554582610654755690700228675818}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 11472936810.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1642:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1642 are not computed
Character table for t16n1642 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.8.137745378125.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
101Data not computed
$149$$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.4.3.3$x^{4} + 298$$4$$1$$3$$C_4$$[\ ]_{4}$
149.8.0.1$x^{8} - x + 11$$1$$8$$0$$C_8$$[\ ]^{8}$