↑

Properties

Label 16.0.81925410828...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 41^{10}$
Root discriminant $41.65$
Ramified primes $5, 41$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21605, -25025, 49765, -32960, 38060, -15125, 14670, -4280, 4091, -717, 548, -154, 120, -64, 18, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 18*x^14 - 64*x^13 + 120*x^12 - 154*x^11 + 548*x^10 - 717*x^9 + 4091*x^8 - 4280*x^7 + 14670*x^6 - 15125*x^5 + 38060*x^4 - 32960*x^3 + 49765*x^2 - 25025*x + 21605)
 
gp: K = bnfinit(x^16 - 2*x^15 + 18*x^14 - 64*x^13 + 120*x^12 - 154*x^11 + 548*x^10 - 717*x^9 + 4091*x^8 - 4280*x^7 + 14670*x^6 - 15125*x^5 + 38060*x^4 - 32960*x^3 + 49765*x^2 - 25025*x + 21605, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 18 x^{14} - 64 x^{13} + 120 x^{12} - 154 x^{11} + 548 x^{10} - 717 x^{9} + 4091 x^{8} - 4280 x^{7} + 14670 x^{6} - 15125 x^{5} + 38060 x^{4} - 32960 x^{3} + 49765 x^{2} - 25025 x + 21605 \)

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:  \(81925410828566900634765625=5^{14}\cdot 41^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.65$
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:  $5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{5}{11} a^{13} - \frac{3}{11} a^{12} + \frac{4}{11} a^{11} - \frac{4}{11} a^{10} - \frac{1}{11} a^{8} + \frac{1}{11} a^{7} - \frac{1}{11} a^{6} - \frac{1}{11} a^{5} + \frac{2}{11} a^{4} - \frac{3}{11} a^{3} + \frac{3}{11} a^{2} - \frac{4}{11} a + \frac{4}{11}$, $\frac{1}{32449835978673530990764954946757629} a^{15} - \frac{890133230891276349278308420759048}{32449835978673530990764954946757629} a^{14} - \frac{470008215924791002768829788516078}{2949985088970320999160450449705239} a^{13} - \frac{6727709502769729505800046351238685}{32449835978673530990764954946757629} a^{12} + \frac{3391631362718932405663770365872519}{32449835978673530990764954946757629} a^{11} - \frac{11794005252335988339310407223411098}{32449835978673530990764954946757629} a^{10} - \frac{9504384386010844619129425898095344}{32449835978673530990764954946757629} a^{9} - \frac{6887883182914701385639684521154838}{32449835978673530990764954946757629} a^{8} + \frac{9237270721166749527326785422105259}{32449835978673530990764954946757629} a^{7} - \frac{7659947171834764273136905706084566}{32449835978673530990764954946757629} a^{6} - \frac{723135985845411405170232757695007}{32449835978673530990764954946757629} a^{5} - \frac{14270571744008224595860154675695212}{32449835978673530990764954946757629} a^{4} + \frac{1445778799686338497624204823098980}{32449835978673530990764954946757629} a^{3} + \frac{6716519644094026617176531685217785}{32449835978673530990764954946757629} a^{2} + \frac{4295121879927764418186324163167985}{32449835978673530990764954946757629} a - \frac{8355295360601972448085521020105008}{32449835978673530990764954946757629}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 249182.839137 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.5125.1, 4.0.210125.1, 4.0.1025.1, 8.4.9051265703125.1, 8.4.5384453125.2, 8.0.44152515625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
41.8.6.2$x^{8} + 943 x^{4} + 242064$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$