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Properties

Label 16.8.13536953436...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{12}\cdot 13^{4}\cdot 29^{6}\cdot 1181^{4}$
Root discriminant $372.15$
Ramified primes $2, 5, 13, 29, 1181$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $(C_2^2\times C_4).C_2^4$ (as 16T471)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![168216722677285441, 23399467424640620, -3913108136820788, 161198981765422, -137019728017313, 21890648688, 624071995234, -11772785364, 17617749905, -158581856, -25846438, 1135436, -319433, 1018, 12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 12*x^14 + 1018*x^13 - 319433*x^12 + 1135436*x^11 - 25846438*x^10 - 158581856*x^9 + 17617749905*x^8 - 11772785364*x^7 + 624071995234*x^6 + 21890648688*x^5 - 137019728017313*x^4 + 161198981765422*x^3 - 3913108136820788*x^2 + 23399467424640620*x + 168216722677285441)
 
gp: K = bnfinit(x^16 - 4*x^15 + 12*x^14 + 1018*x^13 - 319433*x^12 + 1135436*x^11 - 25846438*x^10 - 158581856*x^9 + 17617749905*x^8 - 11772785364*x^7 + 624071995234*x^6 + 21890648688*x^5 - 137019728017313*x^4 + 161198981765422*x^3 - 3913108136820788*x^2 + 23399467424640620*x + 168216722677285441, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 12 x^{14} + 1018 x^{13} - 319433 x^{12} + 1135436 x^{11} - 25846438 x^{10} - 158581856 x^{9} + 17617749905 x^{8} - 11772785364 x^{7} + 624071995234 x^{6} + 21890648688 x^{5} - 137019728017313 x^{4} + 161198981765422 x^{3} - 3913108136820788 x^{2} + 23399467424640620 x + 168216722677285441 \)

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:  \(135369534360064007243398352896000000000000=2^{24}\cdot 5^{12}\cdot 13^{4}\cdot 29^{6}\cdot 1181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $372.15$
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:  $2, 5, 13, 29, 1181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{4}{11} a^{12} + \frac{1}{11} a^{11} - \frac{2}{11} a^{10} + \frac{3}{11} a^{9} - \frac{2}{11} a^{8} - \frac{5}{11} a^{7} - \frac{2}{11} a^{6} - \frac{4}{11} a^{5} + \frac{5}{11} a^{3} + \frac{4}{11} a + \frac{1}{11}$, $\frac{1}{765298131990123410566567814545019018019978988151266169684988800986875665442028825978775567064341886778140335733561223930521641} a^{15} + \frac{1625157207567408337888709960149051436905118233362574690328925779955434450615492618278575282779610090463659992675713477619724}{69572557453647582778778892231365365274543544377387833607726254635170515040184438725343233369485626070740030521232838539138331} a^{14} - \frac{60643983595356850751138596515631116900098861684080249495078085684325935815414359061701138445954946562826930886656306555643285}{765298131990123410566567814545019018019978988151266169684988800986875665442028825978775567064341886778140335733561223930521641} a^{13} + \frac{343222334589347044367113607390811067401072619835417868717260079678989802201486835124405285359030607340142630665254016743625977}{765298131990123410566567814545019018019978988151266169684988800986875665442028825978775567064341886778140335733561223930521641} a^{12} - \frac{264425037159424769737193972313390583515173174507433523009228052249393468644176667801566569511457496546294010071834922919395209}{765298131990123410566567814545019018019978988151266169684988800986875665442028825978775567064341886778140335733561223930521641} a^{11} - \frac{160738320548485873942719349975359139684971553595361197315927043167602266974876275461162361163888081224518787918807824741623055}{765298131990123410566567814545019018019978988151266169684988800986875665442028825978775567064341886778140335733561223930521641} a^{10} + \frac{325087141130100835561295890053842699002717614088468015290681316749568623791401329492616962533736781855101477510832652276209338}{765298131990123410566567814545019018019978988151266169684988800986875665442028825978775567064341886778140335733561223930521641} a^{9} - \frac{135168055914935602175620928365055113272132243772118358736623589380308432008105650104039975634870563506629474599272086659535525}{765298131990123410566567814545019018019978988151266169684988800986875665442028825978775567064341886778140335733561223930521641} a^{8} + \frac{291035466099046430115695625180829924316879101010989219522589284253300551047763829301457651259153108204440472950583466691066911}{765298131990123410566567814545019018019978988151266169684988800986875665442028825978775567064341886778140335733561223930521641} a^{7} + \frac{45088037483653806036860437012599682875128723945583377300494699351450210145555975342263581962342847644522827249896694753257631}{765298131990123410566567814545019018019978988151266169684988800986875665442028825978775567064341886778140335733561223930521641} a^{6} + \frac{21203312285826010033826013765443189425962500018230484039923537508502134137741575840579677968748521302921446098338637232273295}{69572557453647582778778892231365365274543544377387833607726254635170515040184438725343233369485626070740030521232838539138331} a^{5} - \frac{259706821570281301920246068784248444012655197442022356860754504185793654187527353422044960519257406220929679674506334779017735}{765298131990123410566567814545019018019978988151266169684988800986875665442028825978775567064341886778140335733561223930521641} a^{4} + \frac{16607576432975705672548908643086238640114943228031141914374963814037790105913471212779917697501701362996316165697087650852686}{69572557453647582778778892231365365274543544377387833607726254635170515040184438725343233369485626070740030521232838539138331} a^{3} + \frac{341791134819709047006468147153252432315147888625323520893459072320672515781847344590382714374558706786990999115287915219124929}{765298131990123410566567814545019018019978988151266169684988800986875665442028825978775567064341886778140335733561223930521641} a^{2} - \frac{3467979602368401603382193981129237854540639375110576141178709145375266196119572201330236600568532465082569839992434041888827}{765298131990123410566567814545019018019978988151266169684988800986875665442028825978775567064341886778140335733561223930521641} a - \frac{18424090545317788590139347944790946932149812390518430196710982017124895405371860034841197587269903668568690843251215596633590}{69572557453647582778778892231365365274543544377387833607726254635170515040184438725343233369485626070740030521232838539138331}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 46350745858300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4).C_2^4$ (as 16T471):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $(C_2^2\times C_4).C_2^4$
Character table for $(C_2^2\times C_4).C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, \(\Q(\zeta_{20})^+\), 4.4.58000.1, 8.8.3364000000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
1181Data not computed