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Properties

Label 16.8.61400557570...1361.2
Degree $16$
Signature $[8, 4]$
Discriminant $11^{12}\cdot 89^{14}$
Root discriminant $306.73$
Ramified primes $11, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3157789331, -12111231250, 25503742812, -26338087641, 12585895662, -2066647462, -398785415, 218778821, -29351828, -2698673, 1228607, -113391, -7756, 2318, -124, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 124*x^14 + 2318*x^13 - 7756*x^12 - 113391*x^11 + 1228607*x^10 - 2698673*x^9 - 29351828*x^8 + 218778821*x^7 - 398785415*x^6 - 2066647462*x^5 + 12585895662*x^4 - 26338087641*x^3 + 25503742812*x^2 - 12111231250*x + 3157789331)
 
gp: K = bnfinit(x^16 - 3*x^15 - 124*x^14 + 2318*x^13 - 7756*x^12 - 113391*x^11 + 1228607*x^10 - 2698673*x^9 - 29351828*x^8 + 218778821*x^7 - 398785415*x^6 - 2066647462*x^5 + 12585895662*x^4 - 26338087641*x^3 + 25503742812*x^2 - 12111231250*x + 3157789331, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 124 x^{14} + 2318 x^{13} - 7756 x^{12} - 113391 x^{11} + 1228607 x^{10} - 2698673 x^{9} - 29351828 x^{8} + 218778821 x^{7} - 398785415 x^{6} - 2066647462 x^{5} + 12585895662 x^{4} - 26338087641 x^{3} + 25503742812 x^{2} - 12111231250 x + 3157789331 \)

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:  \(6140055757001090562172708710746489091361=11^{12}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $306.73$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{132} a^{12} - \frac{2}{33} a^{11} - \frac{1}{33} a^{10} - \frac{43}{132} a^{9} - \frac{19}{132} a^{8} + \frac{8}{33} a^{7} - \frac{37}{132} a^{6} + \frac{7}{33} a^{5} + \frac{17}{44} a^{4} - \frac{5}{132} a^{3} + \frac{15}{44} a^{2} + \frac{29}{132} a + \frac{4}{33}$, $\frac{1}{132} a^{13} - \frac{1}{66} a^{11} - \frac{3}{44} a^{10} - \frac{1}{4} a^{9} + \frac{1}{11} a^{8} + \frac{7}{44} a^{7} + \frac{31}{66} a^{6} + \frac{1}{12} a^{5} + \frac{7}{132} a^{4} - \frac{61}{132} a^{3} - \frac{7}{132} a^{2} + \frac{25}{66} a - \frac{1}{33}$, $\frac{1}{25608} a^{14} + \frac{3}{2134} a^{13} + \frac{73}{25608} a^{12} - \frac{799}{8536} a^{11} + \frac{193}{4268} a^{10} + \frac{527}{2134} a^{9} + \frac{1337}{8536} a^{8} + \frac{91}{12804} a^{7} + \frac{6299}{25608} a^{6} + \frac{865}{6402} a^{5} - \frac{1141}{6402} a^{4} - \frac{3335}{12804} a^{3} + \frac{1801}{12804} a^{2} - \frac{491}{2328} a - \frac{49}{776}$, $\frac{1}{1265596743226987753013708631599487761249182180728719420042514410184} a^{15} - \frac{5075443062168910080513341881685503359945261969533063166180035}{1265596743226987753013708631599487761249182180728719420042514410184} a^{14} + \frac{1259183915302898883904867459844425445002136926760434537270539899}{1265596743226987753013708631599487761249182180728719420042514410184} a^{13} + \frac{1152677587106025101095546101311189451340943503921228773371301953}{632798371613493876506854315799743880624591090364359710021257205092} a^{12} + \frac{103963575071502732456029995316482369067460608793799177869359485975}{1265596743226987753013708631599487761249182180728719420042514410184} a^{11} + \frac{2500999049860142505840792935754632240448426802620707544865532081}{105466395268915646084475719299957313437431848394059951670209534182} a^{10} + \frac{116713902805098067703473068291678657144396230915802688886086736765}{421865581075662584337902877199829253749727393576239806680838136728} a^{9} + \frac{396673522697815781314067107011410237159048477959214470360329298673}{1265596743226987753013708631599487761249182180728719420042514410184} a^{8} - \frac{95326872118744620249651962675606393040523832615837090303034831503}{1265596743226987753013708631599487761249182180728719420042514410184} a^{7} - \frac{22230636798683267373131233222761322275558832911248537889247074833}{115054249384271613910337148327226160113562016429883583640228582744} a^{6} - \frac{12287101568741290664695666352213480122287120338143196173002833549}{158199592903373469126713578949935970156147772591089927505314301273} a^{5} - \frac{160284196630342467575394941640956231555633166461893171044202770131}{632798371613493876506854315799743880624591090364359710021257205092} a^{4} + \frac{84227453336241472225205585289233436895213212709158399788864437621}{210932790537831292168951438599914626874863696788119903340419068364} a^{3} - \frac{160599775026082566920441293852895540836287291753549603245176624367}{421865581075662584337902877199829253749727393576239806680838136728} a^{2} + \frac{189997089074569339126127733387967511607689734755198668776626227309}{632798371613493876506854315799743880624591090364359710021257205092} a - \frac{310286671163222711366825406522427713601117806246492203902309973767}{1265596743226987753013708631599487761249182180728719420042514410184}$

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:  \( 61373116180800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1, 8.8.647590974205440089.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$11$11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$89$89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$