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Properties

Label 16.0.32051921049...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{10}\cdot 29^{8}$
Root discriminant $25.50$
Ramified primes $3, 5, 29$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $C_2\times C_2^2.D_4$ (as 16T92)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 51, 106, 1773, -289, 983, 649, 108, 326, 108, 59, 38, 4, 11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 11*x^14 + 4*x^13 + 38*x^12 + 59*x^11 + 108*x^10 + 326*x^9 + 108*x^8 + 649*x^7 + 983*x^6 - 289*x^5 + 1773*x^4 + 106*x^3 + 51*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 + 11*x^14 + 4*x^13 + 38*x^12 + 59*x^11 + 108*x^10 + 326*x^9 + 108*x^8 + 649*x^7 + 983*x^6 - 289*x^5 + 1773*x^4 + 106*x^3 + 51*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 11 x^{14} + 4 x^{13} + 38 x^{12} + 59 x^{11} + 108 x^{10} + 326 x^{9} + 108 x^{8} + 649 x^{7} + 983 x^{6} - 289 x^{5} + 1773 x^{4} + 106 x^{3} + 51 x^{2} - 3 x + 1 \)

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:  \(32051921049190634765625=3^{8}\cdot 5^{10}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.50$
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:  $3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{3} a^{7} + \frac{1}{4} a^{6} - \frac{1}{12} a^{4} - \frac{5}{12} a^{3} - \frac{1}{6} a - \frac{1}{4}$, $\frac{1}{12} a^{12} + \frac{1}{6} a^{9} + \frac{5}{12} a^{8} - \frac{1}{12} a^{7} - \frac{1}{12} a^{6} - \frac{1}{12} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{6} a^{2} - \frac{5}{12} a + \frac{1}{12}$, $\frac{1}{12} a^{13} - \frac{1}{6} a^{10} - \frac{1}{4} a^{9} - \frac{1}{12} a^{8} - \frac{1}{12} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{12} a - \frac{1}{3}$, $\frac{1}{92640} a^{14} - \frac{1091}{30880} a^{13} - \frac{79}{18528} a^{12} - \frac{19}{15440} a^{11} + \frac{10027}{92640} a^{10} - \frac{919}{11580} a^{9} - \frac{8243}{92640} a^{8} + \frac{13569}{30880} a^{7} - \frac{9581}{46320} a^{6} - \frac{1316}{2895} a^{5} + \frac{313}{92640} a^{4} - \frac{2299}{5790} a^{3} + \frac{177}{386} a^{2} - \frac{14219}{46320} a - \frac{22691}{92640}$, $\frac{1}{1054223097120} a^{15} - \frac{225077}{131777887140} a^{14} - \frac{6335593259}{263555774280} a^{13} + \frac{19813105811}{1054223097120} a^{12} - \frac{3142801391}{1054223097120} a^{11} - \frac{80299591}{351407699040} a^{10} + \frac{30348583917}{117135899680} a^{9} + \frac{30214298261}{65888943570} a^{8} + \frac{6424485617}{1054223097120} a^{7} - \frac{1635678983}{10335520560} a^{6} + \frac{258368192249}{1054223097120} a^{5} - \frac{284986025783}{1054223097120} a^{4} + \frac{1919915797}{7751640420} a^{3} + \frac{9849332281}{527111548560} a^{2} + \frac{314497393103}{1054223097120} a + \frac{131493980933}{1054223097120}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( \frac{1419407}{22759566} a^{15} - \frac{63525059}{606921760} a^{14} + \frac{389030307}{606921760} a^{13} + \frac{173157959}{364153056} a^{12} + \frac{731201823}{303460880} a^{11} + \frac{2658284167}{606921760} a^{10} + \frac{1760747513}{227595660} a^{9} + \frac{13464998617}{606921760} a^{8} + \frac{7753851407}{606921760} a^{7} + \frac{2205326381}{53551920} a^{6} + \frac{2780342713}{37932610} a^{5} + \frac{56026693}{606921760} a^{4} + \frac{449411151}{4462660} a^{3} + \frac{634939229}{15173044} a^{2} + \frac{861930381}{303460880} a + \frac{1530486307}{1820765280} \) (order $6$)
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:  \( 27482.5974391 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2.D_4$ (as 16T92):

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 $C_2\times C_2^2.D_4$
Character table for $C_2\times C_2^2.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{-435}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{-87}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{29})\), \(\Q(\sqrt{5}, \sqrt{29})\), \(\Q(\sqrt{-15}, \sqrt{29})\), \(\Q(\sqrt{5}, \sqrt{-87})\), \(\Q(\sqrt{-15}, \sqrt{-87})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{145})\), 8.4.19892278125.2, 8.4.19892278125.1, 8.0.35806100625.4

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$