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Properties

Label 16.0.11574317056...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{12}\cdot 41^{4}$
Root discriminant $23.93$
Ramified primes $2, 5, 41$
Class number $2$
Class group $[2]$
Galois group $C_2 \times (C_2^2:C_4)$ (as 16T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, 992, -672, 5272, -5328, 8024, -4176, -855, 876, 432, -366, -48, 72, -3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 3*x^14 + 72*x^13 - 48*x^12 - 366*x^11 + 432*x^10 + 876*x^9 - 855*x^8 - 4176*x^7 + 8024*x^6 - 5328*x^5 + 5272*x^4 - 672*x^3 + 992*x^2 + 16)
 
gp: K = bnfinit(x^16 - 6*x^15 - 3*x^14 + 72*x^13 - 48*x^12 - 366*x^11 + 432*x^10 + 876*x^9 - 855*x^8 - 4176*x^7 + 8024*x^6 - 5328*x^5 + 5272*x^4 - 672*x^3 + 992*x^2 + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 3 x^{14} + 72 x^{13} - 48 x^{12} - 366 x^{11} + 432 x^{10} + 876 x^{9} - 855 x^{8} - 4176 x^{7} + 8024 x^{6} - 5328 x^{5} + 5272 x^{4} - 672 x^{3} + 992 x^{2} + 16 \)

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:  \(11574317056000000000000=2^{24}\cdot 5^{12}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.93$
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 not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{11} + \frac{1}{16} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{8} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{7}{16} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{16} a^{13} + \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{3}{8} a^{8} + \frac{1}{4} a^{6} + \frac{1}{16} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{2624} a^{14} - \frac{1}{1312} a^{13} + \frac{67}{2624} a^{12} + \frac{23}{328} a^{11} + \frac{5}{1312} a^{10} - \frac{203}{1312} a^{9} - \frac{267}{656} a^{8} - \frac{119}{328} a^{7} + \frac{1249}{2624} a^{6} - \frac{319}{656} a^{5} + \frac{315}{1312} a^{4} + \frac{319}{656} a^{2} - \frac{51}{164} a + \frac{21}{328}$, $\frac{1}{340717851990744576064} a^{15} + \frac{12325235641806753}{85179462997686144016} a^{14} - \frac{6475867623314783733}{340717851990744576064} a^{13} + \frac{1232519998352133033}{170358925995372288032} a^{12} + \frac{4707228191851260247}{170358925995372288032} a^{11} - \frac{15838786965125709757}{170358925995372288032} a^{10} + \frac{7518492218703559337}{21294865749421536004} a^{9} - \frac{13573910204446239779}{42589731498843072008} a^{8} - \frac{56079921839945003967}{340717851990744576064} a^{7} - \frac{45894566343771574227}{170358925995372288032} a^{6} + \frac{79053693234219884193}{170358925995372288032} a^{5} + \frac{25069369975235705177}{85179462997686144016} a^{4} - \frac{6622174501341484949}{85179462997686144016} a^{3} - \frac{2923602715674399125}{42589731498843072008} a^{2} + \frac{2823576962946485707}{42589731498843072008} a - \frac{7803798965663418117}{21294865749421536004}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{6727789889287}{3801296992042402} a^{15} + \frac{1293060315775509}{121641503745356864} a^{14} + \frac{260020987738157}{60820751872678432} a^{13} - \frac{14831512444052701}{121641503745356864} a^{12} + \frac{2707947308296061}{30410375936339216} a^{11} + \frac{35383351172705103}{60820751872678432} a^{10} - \frac{44032565312999783}{60820751872678432} a^{9} - \frac{37080847705310177}{30410375936339216} a^{8} + \frac{8323747387826573}{7602593984084804} a^{7} + \frac{19646800221853805}{2966865945008704} a^{6} - \frac{202780899481129203}{15205187968169608} a^{5} + \frac{804840001500438261}{60820751872678432} a^{4} - \frac{131010188313826853}{7602593984084804} a^{3} + \frac{206546282841049203}{30410375936339216} a^{2} - \frac{17683300533000177}{3801296992042402} a + \frac{13694851687425595}{15205187968169608} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 45664.5166523 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:C_4$ (as 16T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2 \times (C_2^2:C_4)$
Character table for $C_2 \times (C_2^2:C_4)$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.1025.1, \(\Q(\zeta_{5})\), 4.0.8000.2, 4.0.65600.4, 4.4.5125.1, 4.4.328000.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.107584000000.4, 8.0.26265625.1, 8.0.107584000000.22, 8.0.107584000000.24, 8.0.107584000000.16, 8.0.4303360000.4, 8.0.64000000.2

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

Sibling fields

Galois closure: data not computed
Degree 16 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$