↑

Properties

Label 16.16.8623412650...6096.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{48}\cdot 3^{12}\cdot 7^{8}$
Root discriminant $48.25$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_8$ (as 16T7)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-383, -3024, -1664, 13344, 11692, -20784, -19736, 14208, 13957, -4272, -4484, 552, 670, -24, -44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 44*x^14 - 24*x^13 + 670*x^12 + 552*x^11 - 4484*x^10 - 4272*x^9 + 13957*x^8 + 14208*x^7 - 19736*x^6 - 20784*x^5 + 11692*x^4 + 13344*x^3 - 1664*x^2 - 3024*x - 383)
 
gp: K = bnfinit(x^16 - 44*x^14 - 24*x^13 + 670*x^12 + 552*x^11 - 4484*x^10 - 4272*x^9 + 13957*x^8 + 14208*x^7 - 19736*x^6 - 20784*x^5 + 11692*x^4 + 13344*x^3 - 1664*x^2 - 3024*x - 383, 1)
 

Normalized defining polynomial

\( x^{16} - 44 x^{14} - 24 x^{13} + 670 x^{12} + 552 x^{11} - 4484 x^{10} - 4272 x^{9} + 13957 x^{8} + 14208 x^{7} - 19736 x^{6} - 20784 x^{5} + 11692 x^{4} + 13344 x^{3} - 1664 x^{2} - 3024 x - 383 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(862341265079199274522116096=2^{48}\cdot 3^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.25$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{44643} a^{14} - \frac{472}{14881} a^{13} + \frac{5984}{44643} a^{12} - \frac{536}{14881} a^{11} - \frac{537}{14881} a^{10} - \frac{2851}{44643} a^{9} + \frac{6917}{44643} a^{8} + \frac{21997}{44643} a^{7} - \frac{3029}{44643} a^{6} - \frac{6728}{14881} a^{5} - \frac{1346}{14881} a^{4} - \frac{9290}{44643} a^{3} + \frac{7357}{14881} a^{2} - \frac{1018}{44643} a - \frac{2550}{14881}$, $\frac{1}{19444043188640883} a^{15} + \frac{164176274981}{19444043188640883} a^{14} + \frac{8206333582618}{845393182114821} a^{13} - \frac{1165820566960172}{19444043188640883} a^{12} + \frac{872808120094585}{6481347729546961} a^{11} - \frac{132743374105792}{6481347729546961} a^{10} + \frac{350902614377081}{6481347729546961} a^{9} + \frac{313563936543974}{19444043188640883} a^{8} + \frac{2677957984189792}{6481347729546961} a^{7} - \frac{3488395589088308}{19444043188640883} a^{6} + \frac{1373174506896534}{6481347729546961} a^{5} - \frac{1812437500756472}{6481347729546961} a^{4} - \frac{1193740332403315}{19444043188640883} a^{3} - \frac{2338899476540452}{19444043188640883} a^{2} - \frac{9027807360059120}{19444043188640883} a + \frac{1910220520545250}{19444043188640883}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 155691205.876 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times Q_8$ (as 16T7):

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 $D_8$
Character table for $D_8$

Intermediate fields

\(\Q(\sqrt{42}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{3}, \sqrt{14})\), \(\Q(\sqrt{6}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{6}, \sqrt{14})\), 8.8.12745506816.1, 8.8.12230590464.1, 8.8.29365647704064.1

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$