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Properties

Label 16.0.59904939404...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 19^{2}\cdot 29^{4}\cdot 31^{2}$
Root discriminant $17.22$
Ramified primes $5, 19, 29, 31$
Class number $2$
Class group $[2]$
Galois group $C_2^4.C_2^3.C_2$ (as 16T547)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -17, 119, -214, 332, -245, 187, -119, 212, -131, 155, -70, 55, -22, 9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 9*x^14 - 22*x^13 + 55*x^12 - 70*x^11 + 155*x^10 - 131*x^9 + 212*x^8 - 119*x^7 + 187*x^6 - 245*x^5 + 332*x^4 - 214*x^3 + 119*x^2 - 17*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 + 9*x^14 - 22*x^13 + 55*x^12 - 70*x^11 + 155*x^10 - 131*x^9 + 212*x^8 - 119*x^7 + 187*x^6 - 245*x^5 + 332*x^4 - 214*x^3 + 119*x^2 - 17*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 9 x^{14} - 22 x^{13} + 55 x^{12} - 70 x^{11} + 155 x^{10} - 131 x^{9} + 212 x^{8} - 119 x^{7} + 187 x^{6} - 245 x^{5} + 332 x^{4} - 214 x^{3} + 119 x^{2} - 17 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(59904939404541015625=5^{12}\cdot 19^{2}\cdot 29^{4}\cdot 31^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.22$
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:  $5, 19, 29, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{1067708899055423973} a^{15} + \frac{121371289740097618}{1067708899055423973} a^{14} - \frac{196945114363203734}{1067708899055423973} a^{13} + \frac{465146598003464539}{1067708899055423973} a^{12} - \frac{333305806973110199}{1067708899055423973} a^{11} - \frac{22174322557654923}{355902966351807991} a^{10} + \frac{203613482965747810}{1067708899055423973} a^{9} - \frac{167309667508412146}{355902966351807991} a^{8} + \frac{400388466389506538}{1067708899055423973} a^{7} - \frac{18373882537678082}{56195205213443367} a^{6} + \frac{43380930750623534}{355902966351807991} a^{5} + \frac{224283594744390238}{1067708899055423973} a^{4} + \frac{8752829564637887}{56195205213443367} a^{3} + \frac{103175187482067840}{355902966351807991} a^{2} - \frac{320990184908307799}{1067708899055423973} a - \frac{494287733880990872}{1067708899055423973}$

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

Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{222313451163117194}{1067708899055423973} a^{15} + \frac{138654742622617686}{355902966351807991} a^{14} - \frac{1942866024871123945}{1067708899055423973} a^{13} + \frac{4637607389808151582}{1067708899055423973} a^{12} - \frac{3867532557614987770}{355902966351807991} a^{11} + \frac{14019361656889221538}{1067708899055423973} a^{10} - \frac{32540116235382824314}{1067708899055423973} a^{9} + \frac{8305430865755697408}{355902966351807991} a^{8} - \frac{43604402961131529331}{1067708899055423973} a^{7} + \frac{1104693781687893965}{56195205213443367} a^{6} - \frac{12863635168743850254}{355902966351807991} a^{5} + \frac{49625119011784435084}{1067708899055423973} a^{4} - \frac{3535344669627331394}{56195205213443367} a^{3} + \frac{12552845279282980975}{355902966351807991} a^{2} - \frac{21518350497435524665}{1067708899055423973} a + \frac{1374173263252052261}{1067708899055423973} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3033.18899555 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T547):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, \(\Q(\zeta_{5})\), 4.0.3625.1, 8.8.309593125.1, 8.0.7739828125.1, 8.0.13140625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
5Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$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$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$