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Properties

Label 16.8.61400557570...1361.1
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![254211683, -920391824, 124007939, 379926932, -54374353, -67333706, 8382561, 5813984, -488858, -433002, 118599, -8568, -2813, 598, -26, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 26*x^14 + 598*x^13 - 2813*x^12 - 8568*x^11 + 118599*x^10 - 433002*x^9 - 488858*x^8 + 5813984*x^7 + 8382561*x^6 - 67333706*x^5 - 54374353*x^4 + 379926932*x^3 + 124007939*x^2 - 920391824*x + 254211683)
 
gp: K = bnfinit(x^16 - 4*x^15 - 26*x^14 + 598*x^13 - 2813*x^12 - 8568*x^11 + 118599*x^10 - 433002*x^9 - 488858*x^8 + 5813984*x^7 + 8382561*x^6 - 67333706*x^5 - 54374353*x^4 + 379926932*x^3 + 124007939*x^2 - 920391824*x + 254211683, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 26 x^{14} + 598 x^{13} - 2813 x^{12} - 8568 x^{11} + 118599 x^{10} - 433002 x^{9} - 488858 x^{8} + 5813984 x^{7} + 8382561 x^{6} - 67333706 x^{5} - 54374353 x^{4} + 379926932 x^{3} + 124007939 x^{2} - 920391824 x + 254211683 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{22} a^{8} - \frac{1}{11} a^{7} - \frac{2}{11} a^{6} - \frac{3}{11} a^{5} + \frac{9}{22} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{22} a^{9} + \frac{3}{22} a^{7} - \frac{3}{22} a^{6} + \frac{4}{11} a^{5} + \frac{7}{22} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{22} a^{10} + \frac{3}{22} a^{7} - \frac{1}{11} a^{6} + \frac{3}{22} a^{5} + \frac{3}{11} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{22} a^{11} + \frac{2}{11} a^{7} + \frac{2}{11} a^{6} - \frac{9}{22} a^{5} - \frac{5}{22} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{88} a^{12} - \frac{1}{88} a^{10} - \frac{1}{88} a^{8} - \frac{1}{4} a^{7} + \frac{13}{88} a^{6} + \frac{1}{4} a^{5} - \frac{9}{44} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{88} a^{13} - \frac{1}{88} a^{11} - \frac{1}{88} a^{9} - \frac{1}{44} a^{8} + \frac{17}{88} a^{7} - \frac{7}{44} a^{6} - \frac{3}{44} a^{5} - \frac{9}{44} a^{4} + \frac{1}{4} a^{2} - \frac{1}{8} a$, $\frac{1}{129712} a^{14} + \frac{51}{129712} a^{13} - \frac{177}{32428} a^{12} + \frac{1973}{129712} a^{11} + \frac{447}{64856} a^{10} + \frac{111}{129712} a^{9} - \frac{227}{64856} a^{8} - \frac{1901}{129712} a^{7} - \frac{21479}{129712} a^{6} - \frac{1069}{5896} a^{5} + \frac{927}{2948} a^{4} - \frac{239}{2948} a^{3} - \frac{1659}{11792} a^{2} - \frac{39}{1072} a - \frac{423}{1072}$, $\frac{1}{163483330897379235180341358497136312495183632} a^{15} + \frac{7761209094961824802667750667784009166}{10217708181086202198771334906071019530948977} a^{14} + \frac{762361380192273933123231338900253345646139}{163483330897379235180341358497136312495183632} a^{13} + \frac{850818193291031741808956391873087615579563}{163483330897379235180341358497136312495183632} a^{12} - \frac{3609857126481265312626660473712721277885961}{163483330897379235180341358497136312495183632} a^{11} - \frac{2655992241787117210498898034270411511139349}{163483330897379235180341358497136312495183632} a^{10} + \frac{2672522934642564490240521815713481528361965}{163483330897379235180341358497136312495183632} a^{9} - \frac{950301640156528260001723366440829947196285}{163483330897379235180341358497136312495183632} a^{8} + \frac{3943603011991244480193002580244408602968907}{40870832724344808795085339624284078123795908} a^{7} - \frac{75919977553236394991512138742839328667853}{14862120990670839561849214408830573863198512} a^{6} + \frac{1706910056089068198278387580692049419392447}{7431060495335419780924607204415286931599256} a^{5} + \frac{65432491964587725148458874774735866600153}{3715530247667709890462303602207643465799628} a^{4} - \frac{39441607059970756286814841548800800332147}{14862120990670839561849214408830573863198512} a^{3} - \frac{101087783532391994044594383898912607027981}{337775477060700899132936691109785769618148} a^{2} + \frac{29478016196345788063398274779053096220939}{61413723101945618024170307474506503566936} a + \frac{21031543646095050088174310520661900704925}{122827446203891236048340614949013007133872}$

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:  \( 62238536126400 \) (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}$