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Properties

Label 16.8.34182052039...7929.4
Degree $16$
Signature $[8, 4]$
Discriminant $37^{14}\cdot 41^{14}$
Root discriminant $607.24$
Ramified primes $37, 41$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1513355408767, -7126041807578, -1937686049708, 2008596388551, -380843328948, -108889517130, 69586907661, -14155155281, 1612862067, -108427352, -1885874, 866668, -80382, 6156, 130, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 130*x^14 + 6156*x^13 - 80382*x^12 + 866668*x^11 - 1885874*x^10 - 108427352*x^9 + 1612862067*x^8 - 14155155281*x^7 + 69586907661*x^6 - 108889517130*x^5 - 380843328948*x^4 + 2008596388551*x^3 - 1937686049708*x^2 - 7126041807578*x + 1513355408767)
 
gp: K = bnfinit(x^16 - x^15 + 130*x^14 + 6156*x^13 - 80382*x^12 + 866668*x^11 - 1885874*x^10 - 108427352*x^9 + 1612862067*x^8 - 14155155281*x^7 + 69586907661*x^6 - 108889517130*x^5 - 380843328948*x^4 + 2008596388551*x^3 - 1937686049708*x^2 - 7126041807578*x + 1513355408767, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 130 x^{14} + 6156 x^{13} - 80382 x^{12} + 866668 x^{11} - 1885874 x^{10} - 108427352 x^{9} + 1612862067 x^{8} - 14155155281 x^{7} + 69586907661 x^{6} - 108889517130 x^{5} - 380843328948 x^{4} + 2008596388551 x^{3} - 1937686049708 x^{2} - 7126041807578 x + 1513355408767 \)

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:  \(341820520390299824133982419608164514148827929=37^{14}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $607.24$
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:  $37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{544703} a^{14} - \frac{210395}{544703} a^{13} - \frac{126160}{544703} a^{12} + \frac{72277}{544703} a^{11} - \frac{6981}{544703} a^{10} + \frac{39697}{544703} a^{9} + \frac{11997}{544703} a^{8} - \frac{254352}{544703} a^{7} - \frac{57583}{544703} a^{6} + \frac{104391}{544703} a^{5} - \frac{67600}{544703} a^{4} + \frac{94052}{544703} a^{3} - \frac{224175}{544703} a^{2} - \frac{232689}{544703} a - \frac{942}{4289}$, $\frac{1}{136599196472143003058424211956209182748143239061702850816577509345157031818129231111359726855137} a^{15} + \frac{11773036710466712048144294323229865979982993056320720200254215900376090515239385241310559}{136599196472143003058424211956209182748143239061702850816577509345157031818129231111359726855137} a^{14} + \frac{22344105819700595393521271006035037686062165976858097902557762711954174867802354542490966017390}{136599196472143003058424211956209182748143239061702850816577509345157031818129231111359726855137} a^{13} + \frac{65003983131677778034440111022739915240609506834502400498486222929092032626769830905536676537140}{136599196472143003058424211956209182748143239061702850816577509345157031818129231111359726855137} a^{12} + \frac{7200404296681833055313809173045561266736413397186125485841157562447540405410251286599666261319}{136599196472143003058424211956209182748143239061702850816577509345157031818129231111359726855137} a^{11} + \frac{22596630708205594463156863635082277949350987686770213243880207897844165070991784450659333544721}{136599196472143003058424211956209182748143239061702850816577509345157031818129231111359726855137} a^{10} - \frac{56314471191644304614222951626631923192384386625715883561740681730577197510831880527252443645565}{136599196472143003058424211956209182748143239061702850816577509345157031818129231111359726855137} a^{9} + \frac{27460646982488698140388631665705090468411473062982253565941728456847356445570679220823913723294}{136599196472143003058424211956209182748143239061702850816577509345157031818129231111359726855137} a^{8} - \frac{32488087493150521013917795605376361039459884843278248022116365894622642815281434700431772976008}{136599196472143003058424211956209182748143239061702850816577509345157031818129231111359726855137} a^{7} + \frac{58743001463550254797590644240490956905762580640118820025052753292625488577427021858286535084988}{136599196472143003058424211956209182748143239061702850816577509345157031818129231111359726855137} a^{6} + \frac{64914147790789970712108665762573978046860508754521080168598121061802823967384187791751407176021}{136599196472143003058424211956209182748143239061702850816577509345157031818129231111359726855137} a^{5} + \frac{38039796897715323861946123872711440776692194739317888445102035954300259964221655882894138424565}{136599196472143003058424211956209182748143239061702850816577509345157031818129231111359726855137} a^{4} + \frac{62593155367471070413081126427855109670245832582325158029792006664812997880930091536002438938519}{136599196472143003058424211956209182748143239061702850816577509345157031818129231111359726855137} a^{3} + \frac{13935712914041330344817539944149153438493849638106141851889346593034206594502153834545438992538}{136599196472143003058424211956209182748143239061702850816577509345157031818129231111359726855137} a^{2} + \frac{54897843419274725233651493576874494543956043227952357721170906486096158201655220592985880949003}{136599196472143003058424211956209182748143239061702850816577509345157031818129231111359726855137} a + \frac{377757976644780886059983174603904339081648488753998079817652438513350473375814287995328617120}{1075584224190102386286804818552828210615301094974038195406122120828008124552198670168186825631}$

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:  $11$
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:  \( 3571872252400000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{1517}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{37}, \sqrt{41})\), 8.8.12187467896636600569.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$41.8.7.2$x^{8} - 1476$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.2$x^{8} - 1476$$8$$1$$7$$C_8$$[\ ]_{8}$