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Properties

Label 16.0.38385422573...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 11^{6}\cdot 31^{6}$
Root discriminant $29.79$
Ramified primes $5, 11, 31$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $(C_2\times C_8):C_2^2$ (as 16T75)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -375, 2390, -5965, 6016, 766, -1610, -2500, 1486, 635, -201, -255, 76, 25, -5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 5*x^14 + 25*x^13 + 76*x^12 - 255*x^11 - 201*x^10 + 635*x^9 + 1486*x^8 - 2500*x^7 - 1610*x^6 + 766*x^5 + 6016*x^4 - 5965*x^3 + 2390*x^2 - 375*x + 25)
 
gp: K = bnfinit(x^16 - 4*x^15 - 5*x^14 + 25*x^13 + 76*x^12 - 255*x^11 - 201*x^10 + 635*x^9 + 1486*x^8 - 2500*x^7 - 1610*x^6 + 766*x^5 + 6016*x^4 - 5965*x^3 + 2390*x^2 - 375*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 5 x^{14} + 25 x^{13} + 76 x^{12} - 255 x^{11} - 201 x^{10} + 635 x^{9} + 1486 x^{8} - 2500 x^{7} - 1610 x^{6} + 766 x^{5} + 6016 x^{4} - 5965 x^{3} + 2390 x^{2} - 375 x + 25 \)

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:  \(383854225736338134765625=5^{12}\cdot 11^{6}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.79$
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:  $5, 11, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{55} a^{14} + \frac{16}{55} a^{13} - \frac{1}{11} a^{12} - \frac{5}{11} a^{11} + \frac{21}{55} a^{10} + \frac{5}{11} a^{9} + \frac{14}{55} a^{8} + \frac{2}{11} a^{7} + \frac{1}{5} a^{6} + \frac{4}{11} a^{5} - \frac{24}{55} a^{3} - \frac{19}{55} a^{2} + \frac{3}{11} a + \frac{5}{11}$, $\frac{1}{77495572544706128117855} a^{15} - \frac{570752169860372044587}{77495572544706128117855} a^{14} - \frac{2146799006247796958363}{77495572544706128117855} a^{13} + \frac{4010030230921181670289}{15499114508941225623571} a^{12} - \frac{9057014109716471978104}{77495572544706128117855} a^{11} + \frac{6097243258728703414647}{77495572544706128117855} a^{10} + \frac{17962347358384301723779}{77495572544706128117855} a^{9} + \frac{5386875011399775354858}{77495572544706128117855} a^{8} - \frac{8856221182549817624644}{77495572544706128117855} a^{7} - \frac{11877205985629881932158}{77495572544706128117855} a^{6} - \frac{7074352635349915542342}{15499114508941225623571} a^{5} + \frac{26485990369043453492341}{77495572544706128117855} a^{4} - \frac{4822314400957297173052}{77495572544706128117855} a^{3} + \frac{16967018376968101279327}{77495572544706128117855} a^{2} - \frac{303640320575782399928}{15499114508941225623571} a - \frac{1364979924513891975843}{15499114508941225623571}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1209954036857862379764}{77495572544706128117855} a^{15} + \frac{910171331994308216915}{15499114508941225623571} a^{14} + \frac{7042055767006392203089}{77495572544706128117855} a^{13} - \frac{5660189669764839189852}{15499114508941225623571} a^{12} - \frac{97999179336480107043199}{77495572544706128117855} a^{11} + \frac{283794223238132080759994}{77495572544706128117855} a^{10} + \frac{302606304021144358824234}{77495572544706128117855} a^{9} - \frac{682540609471113976650404}{77495572544706128117855} a^{8} - \frac{1930111738049408806188019}{77495572544706128117855} a^{7} + \frac{2547453486180782800863229}{77495572544706128117855} a^{6} + \frac{481050177566633930779373}{15499114508941225623571} a^{5} - \frac{296345615815068231006454}{77495572544706128117855} a^{4} - \frac{1432154937737532891266750}{15499114508941225623571} a^{3} + \frac{5655710074704320402947184}{77495572544706128117855} a^{2} - \frac{389446249577194148177675}{15499114508941225623571} a + \frac{32046038376767086652080}{15499114508941225623571} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 199912.820048 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8):C_2^2$ (as 16T75):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times C_8):C_2^2$
Character table for $(C_2\times C_8):C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.8525.1, \(\Q(\zeta_{5})\), 4.0.42625.1, 8.4.619559703125.2, 8.4.24782388125.1, 8.0.1816890625.5

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
31Data not computed