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Properties

Label 16.8.23150163082...5625.2
Degree $16$
Signature $[8, 4]$
Discriminant $5^{14}\cdot 41^{14}$
Root discriminant $105.39$
Ramified primes $5, 41$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-50455, 796850, -3142655, 3221670, 2304100, -6772345, 5011730, -1396365, -54879, 101483, -12232, -4679, 1000, 166, -57, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 57*x^14 + 166*x^13 + 1000*x^12 - 4679*x^11 - 12232*x^10 + 101483*x^9 - 54879*x^8 - 1396365*x^7 + 5011730*x^6 - 6772345*x^5 + 2304100*x^4 + 3221670*x^3 - 3142655*x^2 + 796850*x - 50455)
 
gp: K = bnfinit(x^16 - 2*x^15 - 57*x^14 + 166*x^13 + 1000*x^12 - 4679*x^11 - 12232*x^10 + 101483*x^9 - 54879*x^8 - 1396365*x^7 + 5011730*x^6 - 6772345*x^5 + 2304100*x^4 + 3221670*x^3 - 3142655*x^2 + 796850*x - 50455, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 57 x^{14} + 166 x^{13} + 1000 x^{12} - 4679 x^{11} - 12232 x^{10} + 101483 x^{9} - 54879 x^{8} - 1396365 x^{7} + 5011730 x^{6} - 6772345 x^{5} + 2304100 x^{4} + 3221670 x^{3} - 3142655 x^{2} + 796850 x - 50455 \)

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:  \(231501630828342033704595947265625=5^{14}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.39$
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:  $5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{8}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{9}$, $\frac{1}{123915991178364835709648616031538776147227988620725} a^{15} - \frac{7384053809911971033835069486756860844084822533863}{123915991178364835709648616031538776147227988620725} a^{14} - \frac{8893141164044578806923870453060021011054386247564}{123915991178364835709648616031538776147227988620725} a^{13} - \frac{2171602643450053954985428175431695623519629406487}{24783198235672967141929723206307755229445597724145} a^{12} + \frac{9448693838891114583760726663608155766004943737133}{24783198235672967141929723206307755229445597724145} a^{11} + \frac{1037444425623169059236800180437201849176682472701}{123915991178364835709648616031538776147227988620725} a^{10} + \frac{316359791892297984347446055463349088278798279187}{123915991178364835709648616031538776147227988620725} a^{9} - \frac{61386460244159992474655829632393330288123468314879}{123915991178364835709648616031538776147227988620725} a^{8} - \frac{2697903697722013860366254914657335286934164865702}{24783198235672967141929723206307755229445597724145} a^{7} - \frac{11421871239897639925802383362217946682130578270371}{24783198235672967141929723206307755229445597724145} a^{6} - \frac{2430152720342738625309477467928629276286019780168}{24783198235672967141929723206307755229445597724145} a^{5} + \frac{5049304051508911450188097151245598025823249292439}{24783198235672967141929723206307755229445597724145} a^{4} - \frac{10296345501322851122912544132550856139826348650564}{24783198235672967141929723206307755229445597724145} a^{3} - \frac{3331399323486953503651466925893418240820688011537}{24783198235672967141929723206307755229445597724145} a^{2} - \frac{4539846837217117177740218929273391813614146583299}{24783198235672967141929723206307755229445597724145} a + \frac{9752054644090194483069484310611850532969135339229}{24783198235672967141929723206307755229445597724145}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 3314603613.08 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{205}) \), \(\Q(\sqrt{5}, \sqrt{41})\), 8.8.74220378765625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$41.8.7.2$x^{8} - 1476$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.2$x^{8} - 1476$$8$$1$$7$$C_8$$[\ ]_{8}$