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Properties

Label 16.0.17958563260...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 5^{12}\cdot 7^{12}$
Root discriminant $32.80$
Ramified primes $3, 5, 7$
Class number $32$ (GRH)
Class group $[4, 8]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7795, -8425, 10695, -4920, 1796, -3627, 944, -1492, 2007, -878, 1034, -531, 254, -114, 27, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 27*x^14 - 114*x^13 + 254*x^12 - 531*x^11 + 1034*x^10 - 878*x^9 + 2007*x^8 - 1492*x^7 + 944*x^6 - 3627*x^5 + 1796*x^4 - 4920*x^3 + 10695*x^2 - 8425*x + 7795)
 
gp: K = bnfinit(x^16 - 5*x^15 + 27*x^14 - 114*x^13 + 254*x^12 - 531*x^11 + 1034*x^10 - 878*x^9 + 2007*x^8 - 1492*x^7 + 944*x^6 - 3627*x^5 + 1796*x^4 - 4920*x^3 + 10695*x^2 - 8425*x + 7795, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 27 x^{14} - 114 x^{13} + 254 x^{12} - 531 x^{11} + 1034 x^{10} - 878 x^{9} + 2007 x^{8} - 1492 x^{7} + 944 x^{6} - 3627 x^{5} + 1796 x^{4} - 4920 x^{3} + 10695 x^{2} - 8425 x + 7795 \)

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:  \(1795856326022129150390625=3^{12}\cdot 5^{12}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.80$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10} a^{12} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{2} a^{9} - \frac{2}{5} a^{8} + \frac{3}{10} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{11} + \frac{1}{10} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{3}{10} a^{7} + \frac{2}{5} a^{5} + \frac{3}{10} a^{4} - \frac{1}{2} a$, $\frac{1}{2950} a^{14} - \frac{129}{2950} a^{13} + \frac{42}{1475} a^{12} + \frac{1251}{2950} a^{11} + \frac{429}{2950} a^{10} - \frac{633}{1475} a^{9} - \frac{1013}{2950} a^{8} + \frac{133}{2950} a^{7} + \frac{561}{1475} a^{6} - \frac{807}{2950} a^{5} - \frac{1421}{2950} a^{4} - \frac{14}{59} a^{3} - \frac{137}{590} a^{2} - \frac{141}{590} a + \frac{103}{295}$, $\frac{1}{75625451994302076159821834050} a^{15} + \frac{7397179726960133868029339}{75625451994302076159821834050} a^{14} - \frac{217710800194560752722393093}{75625451994302076159821834050} a^{13} - \frac{663373189579605603158240536}{37812725997151038079910917025} a^{12} + \frac{32607637882668528912858972087}{75625451994302076159821834050} a^{11} - \frac{19312557208569529943668018019}{75625451994302076159821834050} a^{10} + \frac{14043320932391640514146749547}{37812725997151038079910917025} a^{9} - \frac{16088557497369196096251554501}{75625451994302076159821834050} a^{8} + \frac{4437943816946005073030543451}{75625451994302076159821834050} a^{7} + \frac{5952262124318983471922883367}{37812725997151038079910917025} a^{6} + \frac{29425913622848833106945082423}{75625451994302076159821834050} a^{5} - \frac{30782835161555778587057887853}{75625451994302076159821834050} a^{4} + \frac{884442371521452771580248539}{7562545199430207615982183405} a^{3} - \frac{3238134963723597839271553767}{15125090398860415231964366810} a^{2} - \frac{5111022065972376394123141787}{15125090398860415231964366810} a + \frac{6998805927250395598108653563}{15125090398860415231964366810}$

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

Class group and class number

$C_{4}\times C_{8}$, which has order $32$ (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:  \( 14409.2792961 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}, \sqrt{21})\), 4.0.46305.1 x2, 4.0.231525.1 x2, 4.4.6125.1, \(\Q(\zeta_{15})^+\), 8.0.53603825625.5, 8.8.3038765625.1, 8.0.1340095640625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
3Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
7Data not computed