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Properties

Label 16.0.96909452923...000.20
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{12}\cdot 41^{12}$
Root discriminant $364.46$
Ramified primes $2, 5, 41$
Class number $177382400$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 52, 53300]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7233948160000, 0, 3616974080000, 0, 576730928000, 0, 40249864000, 0, 1390691300, 0, 24542600, 0, 214430, 0, 820, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 820*x^14 + 214430*x^12 + 24542600*x^10 + 1390691300*x^8 + 40249864000*x^6 + 576730928000*x^4 + 3616974080000*x^2 + 7233948160000)
 
gp: K = bnfinit(x^16 + 820*x^14 + 214430*x^12 + 24542600*x^10 + 1390691300*x^8 + 40249864000*x^6 + 576730928000*x^4 + 3616974080000*x^2 + 7233948160000, 1)
 

Normalized defining polynomial

\( x^{16} + 820 x^{14} + 214430 x^{12} + 24542600 x^{10} + 1390691300 x^{8} + 40249864000 x^{6} + 576730928000 x^{4} + 3616974080000 x^{2} + 7233948160000 \)

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:  \(96909452923685986042145406976000000000000=2^{44}\cdot 5^{12}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $364.46$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3280=2^{4}\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{3280}(1,·)$, $\chi_{3280}(1157,·)$, $\chi_{3280}(2049,·)$, $\chi_{3280}(329,·)$, $\chi_{3280}(237,·)$, $\chi_{3280}(81,·)$, $\chi_{3280}(1813,·)$, $\chi_{3280}(173,·)$, $\chi_{3280}(409,·)$, $\chi_{3280}(2533,·)$, $\chi_{3280}(1641,·)$, $\chi_{3280}(2797,·)$, $\chi_{3280}(1969,·)$, $\chi_{3280}(1721,·)$, $\chi_{3280}(893,·)$, $\chi_{3280}(1877,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{410} a^{4}$, $\frac{1}{410} a^{5}$, $\frac{1}{410} a^{6}$, $\frac{1}{410} a^{7}$, $\frac{1}{2521500} a^{8} + \frac{1}{1025} a^{6} - \frac{1}{1025} a^{4} - \frac{2}{5} a^{2} + \frac{1}{15}$, $\frac{1}{5043000} a^{9} + \frac{1}{2050} a^{7} + \frac{3}{4100} a^{5} - \frac{1}{5} a^{3} + \frac{1}{30} a$, $\frac{1}{10086000} a^{10} - \frac{7}{8200} a^{6} + \frac{1}{60} a^{2}$, $\frac{1}{20172000} a^{11} - \frac{7}{16400} a^{7} + \frac{1}{120} a^{3} - \frac{1}{2} a$, $\frac{1}{1042085520000} a^{12} + \frac{47}{1270836000} a^{10} - \frac{197}{2541672000} a^{8} + \frac{7}{5904} a^{6} + \frac{3937}{6199200} a^{4} + \frac{1321}{3780} a^{2} + \frac{4}{945}$, $\frac{1}{2084171040000} a^{13} + \frac{47}{2541672000} a^{11} - \frac{197}{5083344000} a^{9} - \frac{37}{59040} a^{7} + \frac{3937}{12398400} a^{5} + \frac{1321}{7560} a^{3} - \frac{941}{1890} a$, $\frac{1}{2630223852480000} a^{14} - \frac{199}{657555963120000} a^{12} - \frac{161717}{6415180128000} a^{10} - \frac{91789}{534598344000} a^{8} + \frac{12479953}{15646780800} a^{6} - \frac{42899}{122240475} a^{4} - \frac{1188731}{2385180} a^{2} + \frac{66373}{198765}$, $\frac{1}{5260447704960000} a^{15} - \frac{199}{1315111926240000} a^{13} - \frac{161717}{12830360256000} a^{11} - \frac{91789}{1069196688000} a^{9} + \frac{12479953}{31293561600} a^{7} - \frac{42899}{244480950} a^{5} + \frac{1196449}{4770360} a^{3} - \frac{66196}{198765} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{52}\times C_{53300}$, which has order $177382400$ (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:  \( 727356.6173138123 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{82}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{410}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{205}) \), \(\Q(\sqrt{2}, \sqrt{41})\), \(\Q(\sqrt{5}, \sqrt{82})\), \(\Q(\sqrt{10}, \sqrt{82})\), \(\Q(\sqrt{10}, \sqrt{41})\), \(\Q(\sqrt{5}, \sqrt{41})\), \(\Q(\sqrt{2}, \sqrt{205})\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.17643776000.4, 4.0.17643776000.6, 4.0.17643776000.7, 4.0.17643776000.8, 8.8.7233948160000.5, 8.0.311302831538176000000.6, 8.0.311302831538176000000.1, 8.0.311302831538176000000.2, 8.0.311302831538176000000.4, 8.0.311302831538176000000.3, 8.0.311302831538176000000.5

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.22.5$x^{8} + 10 x^{4} + 16 x + 36$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.8.22.5$x^{8} + 10 x^{4} + 16 x + 36$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$