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Properties

Label 16.4.59315359635...0369.1
Degree $16$
Signature $[4, 6]$
Discriminant $17^{14}\cdot 137^{4}$
Root discriminant $40.82$
Ramified primes $17, 137$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![104464, -61232, 92888, 8314, -10433, -9687, -5008, -596, 130, 2113, -590, 305, -156, -14, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 14*x^13 - 156*x^12 + 305*x^11 - 590*x^10 + 2113*x^9 + 130*x^8 - 596*x^7 - 5008*x^6 - 9687*x^5 - 10433*x^4 + 8314*x^3 + 92888*x^2 - 61232*x + 104464)
 
gp: K = bnfinit(x^16 - 3*x^15 - 14*x^13 - 156*x^12 + 305*x^11 - 590*x^10 + 2113*x^9 + 130*x^8 - 596*x^7 - 5008*x^6 - 9687*x^5 - 10433*x^4 + 8314*x^3 + 92888*x^2 - 61232*x + 104464, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 14 x^{13} - 156 x^{12} + 305 x^{11} - 590 x^{10} + 2113 x^{9} + 130 x^{8} - 596 x^{7} - 5008 x^{6} - 9687 x^{5} - 10433 x^{4} + 8314 x^{3} + 92888 x^{2} - 61232 x + 104464 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(59315359635608350207210369=17^{14}\cdot 137^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.82$
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:  $17, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} + \frac{3}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{14} + \frac{1}{32} a^{13} + \frac{1}{32} a^{12} + \frac{7}{32} a^{11} - \frac{7}{32} a^{10} - \frac{1}{8} a^{9} - \frac{5}{32} a^{8} - \frac{11}{32} a^{7} - \frac{3}{32} a^{6} + \frac{13}{32} a^{5} - \frac{7}{32} a^{4} + \frac{1}{16} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{3206637763783464340874828811039137344} a^{15} - \frac{22597520765386817665785064353139785}{1603318881891732170437414405519568672} a^{14} - \frac{56522960635788902059669766802178711}{1603318881891732170437414405519568672} a^{13} - \frac{24603048872669794840745020124217457}{200414860236466521304676800689946084} a^{12} - \frac{20195339254682278042915752337782217}{200414860236466521304676800689946084} a^{11} - \frac{52832443188483992483604595806927563}{3206637763783464340874828811039137344} a^{10} + \frac{345861224934240814382865714772646243}{3206637763783464340874828811039137344} a^{9} + \frac{26622194058964598771627632542283303}{801659440945866085218707202759784336} a^{8} + \frac{31061331474712391634804755292005037}{1603318881891732170437414405519568672} a^{7} - \frac{85253393241586944186034263566722919}{1603318881891732170437414405519568672} a^{6} + \frac{560967851739170099195789971937294807}{1603318881891732170437414405519568672} a^{5} + \frac{1162305783456543473025066754390544515}{3206637763783464340874828811039137344} a^{4} - \frac{751075861820985047586613274071538985}{1603318881891732170437414405519568672} a^{3} + \frac{73112634626005313880331419406774217}{400829720472933042609353601379892168} a^{2} + \frac{17279860323154851539513098966502380}{50103715059116630326169200172486521} a - \frac{52112480068248124715378931190712565}{200414860236466521304676800689946084}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  $9$
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:  \( 516481.842312 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 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

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
137Data not computed