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Properties

Label 16.8.31087867512...0961.2
Degree $16$
Signature $[8, 4]$
Discriminant $31^{10}\cdot 41^{14}$
Root discriminant $220.44$
Ramified primes $31, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2586361, 13122331, -22542750, 11362659, 8819154, -11779338, 1809688, 2910067, -1154321, -229221, 148983, 1684, -5235, 288, -7, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 7*x^14 + 288*x^13 - 5235*x^12 + 1684*x^11 + 148983*x^10 - 229221*x^9 - 1154321*x^8 + 2910067*x^7 + 1809688*x^6 - 11779338*x^5 + 8819154*x^4 + 11362659*x^3 - 22542750*x^2 + 13122331*x - 2586361)
 
gp: K = bnfinit(x^16 - 6*x^15 - 7*x^14 + 288*x^13 - 5235*x^12 + 1684*x^11 + 148983*x^10 - 229221*x^9 - 1154321*x^8 + 2910067*x^7 + 1809688*x^6 - 11779338*x^5 + 8819154*x^4 + 11362659*x^3 - 22542750*x^2 + 13122331*x - 2586361, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 7 x^{14} + 288 x^{13} - 5235 x^{12} + 1684 x^{11} + 148983 x^{10} - 229221 x^{9} - 1154321 x^{8} + 2910067 x^{7} + 1809688 x^{6} - 11779338 x^{5} + 8819154 x^{4} + 11362659 x^{3} - 22542750 x^{2} + 13122331 x - 2586361 \)

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:  \(31087867512274251337747808514764310961=31^{10}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $220.44$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{92} a^{14} - \frac{1}{23} a^{13} - \frac{1}{46} a^{11} + \frac{5}{92} a^{10} - \frac{19}{46} a^{9} - \frac{3}{23} a^{8} - \frac{45}{92} a^{7} - \frac{37}{92} a^{6} + \frac{1}{46} a^{5} - \frac{43}{92} a^{4} + \frac{5}{46} a^{3} + \frac{7}{92} a^{2} - \frac{5}{92} a + \frac{13}{92}$, $\frac{1}{15763916968637724879282058847785028787764829091256} a^{15} - \frac{19534493959942732082843452814461000998877794269}{15763916968637724879282058847785028787764829091256} a^{14} + \frac{910210029431829530798418099839086745962449957113}{3940979242159431219820514711946257196941207272814} a^{13} + \frac{514727762053030179556243936431088762292785874027}{3940979242159431219820514711946257196941207272814} a^{12} + \frac{3512501250232785672410710589100579456121979549021}{15763916968637724879282058847785028787764829091256} a^{11} + \frac{4674954969142412843927781257024566835982963747925}{15763916968637724879282058847785028787764829091256} a^{10} - \frac{1003327939218853263823641266892554831756959365545}{3940979242159431219820514711946257196941207272814} a^{9} - \frac{2798366334187085245635002973936261472030097618869}{15763916968637724879282058847785028787764829091256} a^{8} + \frac{1932289732637132367114643809490163866855223790357}{7881958484318862439641029423892514393882414545628} a^{7} + \frac{3870771508217218010957163786117337018443503793233}{15763916968637724879282058847785028787764829091256} a^{6} + \frac{4874714216853004050415794640988703945509316402469}{15763916968637724879282058847785028787764829091256} a^{5} + \frac{140539442319493705619169688394245639004753223737}{685387694288596733881828645555870816859340395272} a^{4} - \frac{2200745487916050811533630986531328760497191814347}{15763916968637724879282058847785028787764829091256} a^{3} + \frac{531582084226699082466065629822787657355655389945}{1970489621079715609910257355973128598470603636407} a^{2} - \frac{3145553591070905377807182769497969329602132077607}{7881958484318862439641029423892514393882414545628} a + \frac{6863993392732649167543386124622896886522539948781}{15763916968637724879282058847785028787764829091256}$

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:  \( 4136587602960 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.179859661768855001.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$31$31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
41Data not computed