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Properties

Label 16.8.35717732754...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{56}\cdot 5^{6}\cdot 17^{2}\cdot 41^{6}\cdot 137^{2}\cdot 35089^{2}$
Root discriminant $812.00$
Ramified primes $2, 5, 17, 41, 137, 35089$
Class number $16$ (GRH)
Class group $[2, 2, 2, 2]$ (GRH)
Galois group 16T1127

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4490644386843206976400, 0, -244559727285312582720, 0, 521183847139500304, 0, 2977885992852704, 0, -7383828560092, 0, -3440773616, 0, 8466180, 0, 6016, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 6016*x^14 + 8466180*x^12 - 3440773616*x^10 - 7383828560092*x^8 + 2977885992852704*x^6 + 521183847139500304*x^4 - 244559727285312582720*x^2 + 4490644386843206976400)
 
gp: K = bnfinit(x^16 + 6016*x^14 + 8466180*x^12 - 3440773616*x^10 - 7383828560092*x^8 + 2977885992852704*x^6 + 521183847139500304*x^4 - 244559727285312582720*x^2 + 4490644386843206976400, 1)
 

Normalized defining polynomial

\( x^{16} + 6016 x^{14} + 8466180 x^{12} - 3440773616 x^{10} - 7383828560092 x^{8} + 2977885992852704 x^{6} + 521183847139500304 x^{4} - 244559727285312582720 x^{2} + 4490644386843206976400 \)

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:  \(35717732754345647446813752640158274945024000000=2^{56}\cdot 5^{6}\cdot 17^{2}\cdot 41^{6}\cdot 137^{2}\cdot 35089^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $812.00$
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:  $2, 5, 17, 41, 137, 35089$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{6} a^{6} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{7} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{36} a^{8} + \frac{1}{18} a^{6} - \frac{2}{9} a^{4} - \frac{1}{9}$, $\frac{1}{72} a^{9} + \frac{1}{36} a^{7} + \frac{5}{36} a^{5} - \frac{1}{2} a^{3} - \frac{1}{18} a$, $\frac{1}{72} a^{10} - \frac{1}{12} a^{6} + \frac{2}{9} a^{4} + \frac{5}{18} a^{2} + \frac{4}{9}$, $\frac{1}{72} a^{11} - \frac{1}{12} a^{7} + \frac{2}{9} a^{5} + \frac{5}{18} a^{3} + \frac{4}{9} a$, $\frac{1}{72} a^{12} + \frac{1}{18} a^{6} + \frac{1}{9} a^{4} + \frac{1}{9} a^{2} + \frac{1}{3}$, $\frac{1}{144} a^{13} + \frac{1}{36} a^{7} + \frac{1}{18} a^{5} - \frac{4}{9} a^{3} + \frac{1}{6} a$, $\frac{1}{14711461750469534530516795381482101074053909118300310348508303596730942000} a^{14} + \frac{27689862510081829424385426912705924697376104026886253383528534536922603}{7355730875234767265258397690741050537026954559150155174254151798365471000} a^{12} + \frac{1761071545947411765998912078158489837358542270148708651707121257631791}{735573087523476726525839769074105053702695455915015517425415179836547100} a^{10} - \frac{61999957516146292425622777465161644882297433875164397777203552728709}{12019168096788835400749015834544200223900252547630972506951228428701750} a^{8} + \frac{4487174917830560430917756464750462391420709527067645010758219392705119}{54086256435549759303370571255448901007551136464339376281280527929157875} a^{6} + \frac{209445471509196387283591060584654685365386498688151040416711168583165453}{1838932718808691816314599422685262634256738639787538793563537949591367750} a^{4} - \frac{18921620470811307921545303088318482277742742831432460276760457468668517}{1838932718808691816314599422685262634256738639787538793563537949591367750} a^{2} - \frac{5016078407804134528405694749324356530008344175153314639679342}{54883462604181731776477225726722739929417799295854354476698775}$, $\frac{1}{14711461750469534530516795381482101074053909118300310348508303596730942000} a^{15} - \frac{46783203802541442057595780768213852508399938601090759542028372570119669}{14711461750469534530516795381482101074053909118300310348508303596730942000} a^{13} + \frac{1761071545947411765998912078158489837358542270148708651707121257631791}{735573087523476726525839769074105053702695455915015517425415179836547100} a^{11} - \frac{61999957516146292425622777465161644882297433875164397777203552728709}{12019168096788835400749015834544200223900252547630972506951228428701750} a^{9} + \frac{11939115622927824023296517941729749453732711834455093789557263356469601}{216345025742199037213482285021795604030204545857357505125122111716631500} a^{7} + \frac{53641271343245643188612212995514491731117176016643887053812863469600289}{919466359404345908157299711342631317128369319893769396781768974795683875} a^{5} + \frac{798381810110829499329387773660687132947474430407473670195923075683050483}{1838932718808691816314599422685262634256738639787538793563537949591367750} a^{3} - \frac{28326644350335512982303798074222959703155954782258080771591609}{109766925208363463552954451453445479858835598591708708953397550} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (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:  \( 35248379797500000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1127:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 58 conjugacy class representatives for t16n1127 are not computed
Character table for t16n1127 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.44066406400.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$41$41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$137$$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
137.4.2.2$x^{4} - 137 x^{2} + 112614$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
137.8.0.1$x^{8} - x + 5$$1$$8$$0$$C_8$$[\ ]^{8}$
35089Data not computed