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Properties

Label 16.8.12054181652...3369.1
Degree $16$
Signature $[8, 4]$
Discriminant $61^{14}\cdot 73^{14}$
Root discriminant $1558.01$
Ramified primes $61, 73$
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![-18530020609680400767, 1049166479823709608, 593366179212502821, -34592170126209414, -6666671433405269, 426501635882804, 31617335805990, -2778720508658, -33442665892, 9762162282, -177549808, -18427412, 649621, 16767, -1151, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 1151*x^14 + 16767*x^13 + 649621*x^12 - 18427412*x^11 - 177549808*x^10 + 9762162282*x^9 - 33442665892*x^8 - 2778720508658*x^7 + 31617335805990*x^6 + 426501635882804*x^5 - 6666671433405269*x^4 - 34592170126209414*x^3 + 593366179212502821*x^2 + 1049166479823709608*x - 18530020609680400767)
 
gp: K = bnfinit(x^16 - 2*x^15 - 1151*x^14 + 16767*x^13 + 649621*x^12 - 18427412*x^11 - 177549808*x^10 + 9762162282*x^9 - 33442665892*x^8 - 2778720508658*x^7 + 31617335805990*x^6 + 426501635882804*x^5 - 6666671433405269*x^4 - 34592170126209414*x^3 + 593366179212502821*x^2 + 1049166479823709608*x - 18530020609680400767, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 1151 x^{14} + 16767 x^{13} + 649621 x^{12} - 18427412 x^{11} - 177549808 x^{10} + 9762162282 x^{9} - 33442665892 x^{8} - 2778720508658 x^{7} + 31617335805990 x^{6} + 426501635882804 x^{5} - 6666671433405269 x^{4} - 34592170126209414 x^{3} + 593366179212502821 x^{2} + 1049166479823709608 x - 18530020609680400767 \)

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:  \(1205418165202307934078936492069912344967184840483369=61^{14}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1558.01$
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:  $61, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{171} a^{12} + \frac{17}{171} a^{11} + \frac{23}{171} a^{10} - \frac{26}{57} a^{9} - \frac{28}{171} a^{8} - \frac{2}{19} a^{7} + \frac{7}{19} a^{6} - \frac{25}{57} a^{5} - \frac{28}{171} a^{4} + \frac{1}{19} a^{3} + \frac{1}{171} a^{2} - \frac{16}{57} a + \frac{1}{19}$, $\frac{1}{171} a^{13} + \frac{1}{9} a^{11} - \frac{13}{171} a^{10} + \frac{44}{171} a^{9} + \frac{2}{171} a^{8} + \frac{3}{19} a^{7} + \frac{17}{57} a^{6} + \frac{50}{171} a^{5} - \frac{28}{171} a^{4} + \frac{4}{9} a^{3} + \frac{49}{171} a^{2} - \frac{10}{57} a + \frac{2}{19}$, $\frac{1}{9747} a^{14} - \frac{7}{3249} a^{13} + \frac{25}{9747} a^{12} - \frac{25}{9747} a^{11} + \frac{227}{9747} a^{10} - \frac{3328}{9747} a^{9} + \frac{442}{1083} a^{8} - \frac{1519}{3249} a^{7} - \frac{643}{9747} a^{6} + \frac{3602}{9747} a^{5} - \frac{188}{9747} a^{4} + \frac{3352}{9747} a^{3} + \frac{1169}{3249} a^{2} - \frac{31}{361} a - \frac{107}{361}$, $\frac{1}{1409375202702979513815598743010468329677437997996826078527746428236359763680633048644456549356771603264903673226555810971} a^{15} - \frac{6562039038441777374182550248762438598363315764581125710328070045256707828366974135778041289998086655415106288955687}{156597244744775501535066527001163147741937555332980675391971825359595529297848116516050727706307955918322630358506201219} a^{14} - \frac{818974716693931849300053668107164677876756797107827894419520952889234917958064614340642422520957789596203452305546010}{1409375202702979513815598743010468329677437997996826078527746428236359763680633048644456549356771603264903673226555810971} a^{13} - \frac{1004501599565128214373660776513645929620505015349234782447588704978884318941950147600867158338711545998097429415721086}{1409375202702979513815598743010468329677437997996826078527746428236359763680633048644456549356771603264903673226555810971} a^{12} - \frac{24199071443503106982570929458373181558359619565088136392922288970770572679076622198388314723784240110142472119218017300}{1409375202702979513815598743010468329677437997996826078527746428236359763680633048644456549356771603264903673226555810971} a^{11} + \frac{175740730737986558335698831932384544637780826033095314785177505375843208611660148488005075365514955911917262952004734239}{1409375202702979513815598743010468329677437997996826078527746428236359763680633048644456549356771603264903673226555810971} a^{10} + \frac{216465218640176011882411182050578147359391769282395860852949474201309621710120893845167217066337399723860135727012525321}{469791734234326504605199581003489443225812665998942026175915476078786587893544349548152183118923867754967891075518603657} a^{9} - \frac{602012056194997740855558897309859583122608422165755516206970812652137625828812951634927860272219005176758081842227694}{469791734234326504605199581003489443225812665998942026175915476078786587893544349548152183118923867754967891075518603657} a^{8} - \frac{272023086374402350959922491504007697412760980068086999762456684539929881936440604600372057609354499829660293289035401642}{1409375202702979513815598743010468329677437997996826078527746428236359763680633048644456549356771603264903673226555810971} a^{7} - \frac{113823078666533513582925187340148289425454963110588989237361853787450683360370906312546731152696036993877754792260771975}{1409375202702979513815598743010468329677437997996826078527746428236359763680633048644456549356771603264903673226555810971} a^{6} + \frac{476142802858954305526549385837910647171362420920005532792124970534022360883954301753976991981267374591742711473659261950}{1409375202702979513815598743010468329677437997996826078527746428236359763680633048644456549356771603264903673226555810971} a^{5} - \frac{322680563601876813498952621696361467144937492145285383018306275705886714085313791435481629866525443803269916314228645705}{1409375202702979513815598743010468329677437997996826078527746428236359763680633048644456549356771603264903673226555810971} a^{4} - \frac{73314782793489624231772561508996728633913630401777367525906861386322249639193097882525173879247816667101181451047679875}{156597244744775501535066527001163147741937555332980675391971825359595529297848116516050727706307955918322630358506201219} a^{3} - \frac{68668164182437994457248382594217035608586254132476572247272024274670058244093972322386901858834551803243296003654234492}{156597244744775501535066527001163147741937555332980675391971825359595529297848116516050727706307955918322630358506201219} a^{2} + \frac{16096887264144079732265612458958047155709397214439318090988156085761460903702891424221627783890736769184429550641113870}{156597244744775501535066527001163147741937555332980675391971825359595529297848116516050727706307955918322630358506201219} a + \frac{16827999685707244271273128306283851195291211782143840047866353074991977786489080936223750399869764686822090678178078023}{52199081581591833845022175667054382580645851777660225130657275119865176432616038838683575902102651972774210119502067073}$

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:  \( 10563473953900000000 \) (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

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{4453}) \), \(\Q(\sqrt{73}) \), \(\Q(\sqrt{61}, \sqrt{73})\), 8.8.7796795992041567776329.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$61$61.8.7.1$x^{8} - 61$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.1$x^{8} - 61$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$73$73.8.7.4$x^{8} - 1140625$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.4$x^{8} - 1140625$$8$$1$$7$$C_8$$[\ ]_{8}$