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Properties

Label 16.8.23563276747...0553.3
Degree $16$
Signature $[8, 4]$
Discriminant $61^{2}\cdot 97^{15}$
Root discriminant $121.83$
Ramified primes $61, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1223

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![119309, 313014, -1124560, -876464, 1214523, -376128, 224890, -144545, 47818, -11172, 3461, 860, -1244, 149, 26, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 26*x^14 + 149*x^13 - 1244*x^12 + 860*x^11 + 3461*x^10 - 11172*x^9 + 47818*x^8 - 144545*x^7 + 224890*x^6 - 376128*x^5 + 1214523*x^4 - 876464*x^3 - 1124560*x^2 + 313014*x + 119309)
 
gp: K = bnfinit(x^16 - 7*x^15 + 26*x^14 + 149*x^13 - 1244*x^12 + 860*x^11 + 3461*x^10 - 11172*x^9 + 47818*x^8 - 144545*x^7 + 224890*x^6 - 376128*x^5 + 1214523*x^4 - 876464*x^3 - 1124560*x^2 + 313014*x + 119309, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 26 x^{14} + 149 x^{13} - 1244 x^{12} + 860 x^{11} + 3461 x^{10} - 11172 x^{9} + 47818 x^{8} - 144545 x^{7} + 224890 x^{6} - 376128 x^{5} + 1214523 x^{4} - 876464 x^{3} - 1124560 x^{2} + 313014 x + 119309 \)

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:  \(2356327674777993305467029827040553=61^{2}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $121.83$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{59768100359694797356120577215665422697782182974662} a^{15} + \frac{632663462696634745646176805668645548891673134225}{29884050179847398678060288607832711348891091487331} a^{14} + \frac{3738283865442191962587558605000230290879952387773}{59768100359694797356120577215665422697782182974662} a^{13} + \frac{6283231302512351506926343213685971797385092132475}{29884050179847398678060288607832711348891091487331} a^{12} - \frac{12136974181922215896392746614615879335220849757446}{29884050179847398678060288607832711348891091487331} a^{11} - \frac{7715187570344156755486862187313354614028706814459}{59768100359694797356120577215665422697782182974662} a^{10} + \frac{810490837465459545687595939213498788100265099338}{29884050179847398678060288607832711348891091487331} a^{9} + \frac{7920857978714027791007737834934677255477168842868}{29884050179847398678060288607832711348891091487331} a^{8} + \frac{10642598715038892257812726708272829739765221944287}{59768100359694797356120577215665422697782182974662} a^{7} - \frac{93346211918149439926667303086192535119502163077}{694977911159241829722332293205411891834676546217} a^{6} + \frac{40431579826395274413935257060781558454463895289}{694977911159241829722332293205411891834676546217} a^{5} - \frac{19644949732324287096860088448688717857214698480959}{59768100359694797356120577215665422697782182974662} a^{4} - \frac{13993971144463167192727551941722862536633009085529}{29884050179847398678060288607832711348891091487331} a^{3} + \frac{7040313354855952566602552186785129057574269873509}{29884050179847398678060288607832711348891091487331} a^{2} - \frac{27349189687399803803483404581577610204259507634505}{59768100359694797356120577215665422697782182974662} a - \frac{15860955235589254996835457677009914674354451961}{114718042916880609128830282563657241262537779222}$

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:  \( 24755827843.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1223:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1223
Character table for t16n1223 is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ $16$

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
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
97Data not computed