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Properties

Label 16.0.26268600651...00.134
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 17^{12}$
Root discriminant $387.90$
Ramified primes $2, 3, 5, 17$
Class number $230400000$ (GRH)
Class group $[2, 20, 20, 20, 120, 120]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![50975958665025, 0, 29152604149200, 0, 4151692426860, 0, 195172069320, 0, 4120428771, 0, 45194976, 0, 268056, 0, 816, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 816*x^14 + 268056*x^12 + 45194976*x^10 + 4120428771*x^8 + 195172069320*x^6 + 4151692426860*x^4 + 29152604149200*x^2 + 50975958665025)
 
gp: K = bnfinit(x^16 + 816*x^14 + 268056*x^12 + 45194976*x^10 + 4120428771*x^8 + 195172069320*x^6 + 4151692426860*x^4 + 29152604149200*x^2 + 50975958665025, 1)
 

Normalized defining polynomial

\( x^{16} + 816 x^{14} + 268056 x^{12} + 45194976 x^{10} + 4120428771 x^{8} + 195172069320 x^{6} + 4151692426860 x^{4} + 29152604149200 x^{2} + 50975958665025 \)

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:  \(262686006513622818702917369856000000000000=2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $387.90$
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:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4080=2^{4}\cdot 3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{4080}(1,·)$, $\chi_{4080}(1801,·)$, $\chi_{4080}(3149,·)$, $\chi_{4080}(2449,·)$, $\chi_{4080}(4067,·)$, $\chi_{4080}(149,·)$, $\chi_{4080}(1883,·)$, $\chi_{4080}(2143,·)$, $\chi_{4080}(803,·)$, $\chi_{4080}(3943,·)$, $\chi_{4080}(169,·)$, $\chi_{4080}(1067,·)$, $\chi_{4080}(1327,·)$, $\chi_{4080}(1781,·)$, $\chi_{4080}(3127,·)$, $\chi_{4080}(701,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{153} a^{4}$, $\frac{1}{153} a^{5}$, $\frac{1}{459} a^{6}$, $\frac{1}{459} a^{7}$, $\frac{1}{351135} a^{8} - \frac{7}{6885} a^{6} + \frac{2}{2295} a^{4} - \frac{1}{3}$, $\frac{1}{351135} a^{9} - \frac{7}{6885} a^{7} + \frac{2}{2295} a^{5} - \frac{1}{3} a$, $\frac{1}{251763795} a^{10} - \frac{23}{27973755} a^{8} + \frac{13}{182835} a^{6} - \frac{217}{109701} a^{4} + \frac{11}{2151} a^{2} + \frac{340}{717}$, $\frac{1}{76787957475} a^{11} - \frac{1457}{8531995275} a^{9} + \frac{108067}{167294025} a^{7} + \frac{481456}{167294025} a^{5} - \frac{24367}{656055} a^{3} - \frac{27623}{218685} a$, $\frac{1}{3916185831225} a^{12} + \frac{4}{25595985825} a^{10} + \frac{5476}{25595985825} a^{8} + \frac{38368}{501882075} a^{6} - \frac{16154}{11152935} a^{4} + \frac{104246}{656055} a^{2} + \frac{61}{717}$, $\frac{1}{3916185831225} a^{13} - \frac{1663}{2843998425} a^{9} - \frac{61504}{501882075} a^{7} + \frac{23234}{9840825} a^{5} - \frac{8144}{131211} a^{3} - \frac{4798}{72895} a$, $\frac{1}{11748557493675} a^{14} - \frac{22}{76787957475} a^{10} - \frac{35713}{25595985825} a^{8} - \frac{38734}{167294025} a^{6} - \frac{19097}{11152935} a^{4} - \frac{68684}{656055} a^{2} - \frac{70}{717}$, $\frac{1}{11748557493675} a^{15} + \frac{2783}{5119197165} a^{9} + \frac{1733}{1645515} a^{7} - \frac{19829}{9840825} a^{5} + \frac{17099}{218685} a^{3} + \frac{33298}{72895} a$

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

Class group and class number

$C_{2}\times C_{20}\times C_{20}\times C_{20}\times C_{120}\times C_{120}$, which has order $230400000$ (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:  \( -1 \) (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:  \( 663170.015206084 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{170}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{5}) \), 4.0.11319552000.2, \(\Q(\sqrt{5}, \sqrt{34})\), 4.0.11319552000.7, 4.0.2263910400.16, 4.0.90556416.4, 4.4.2312000.1, \(\Q(\zeta_{20})^+\), 8.0.128132257480704000000.37, 8.0.5125290299228160000.22, 8.8.85525504000000.2

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$
17Data not computed