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Properties

Label 16.0.83558672988...3376.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 7^{6}\cdot 17^{2}\cdot 73^{2}$
Root discriminant $74.15$
Ramified primes $2, 7, 17, 73$
Class number $11568$ (GRH)
Class group $[2, 2, 2, 1446]$ (GRH)
Galois group 16T1086

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16366712, -13908224, 26002080, -14357632, 13795312, -4535584, 3274368, -600224, 458360, -36672, 41104, -592, 2224, 0, 64, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 64*x^14 + 2224*x^12 - 592*x^11 + 41104*x^10 - 36672*x^9 + 458360*x^8 - 600224*x^7 + 3274368*x^6 - 4535584*x^5 + 13795312*x^4 - 14357632*x^3 + 26002080*x^2 - 13908224*x + 16366712)
 
gp: K = bnfinit(x^16 + 64*x^14 + 2224*x^12 - 592*x^11 + 41104*x^10 - 36672*x^9 + 458360*x^8 - 600224*x^7 + 3274368*x^6 - 4535584*x^5 + 13795312*x^4 - 14357632*x^3 + 26002080*x^2 - 13908224*x + 16366712, 1)
 

Normalized defining polynomial

\( x^{16} + 64 x^{14} + 2224 x^{12} - 592 x^{11} + 41104 x^{10} - 36672 x^{9} + 458360 x^{8} - 600224 x^{7} + 3274368 x^{6} - 4535584 x^{5} + 13795312 x^{4} - 14357632 x^{3} + 26002080 x^{2} - 13908224 x + 16366712 \)

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:  \(835586729888343128721772773376=2^{62}\cdot 7^{6}\cdot 17^{2}\cdot 73^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.15$
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:  $2, 7, 17, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{4} a^{13}$, $\frac{1}{28} a^{14} - \frac{1}{14} a^{13} - \frac{3}{28} a^{12} + \frac{1}{28} a^{11} + \frac{3}{14} a^{10} + \frac{1}{7} a^{9} + \frac{1}{7} a^{7} + \frac{3}{14} a^{6} - \frac{1}{7} a^{5} + \frac{2}{7} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2} - \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{33445763835484077513345662047861319791757573756} a^{15} - \frac{177941839548243027927796892451497633677799973}{33445763835484077513345662047861319791757573756} a^{14} - \frac{525024696834364267757020821641529947488030357}{8361440958871019378336415511965329947939393439} a^{13} + \frac{649390731634005329211163065537159501561138057}{33445763835484077513345662047861319791757573756} a^{12} + \frac{290329350382469028657597281067284785482994637}{4777966262212011073335094578265902827393939108} a^{11} + \frac{1331649510612712198320010133535241178389645617}{16722881917742038756672831023930659895878786878} a^{10} + \frac{551760585769669836329802512377080251026072241}{8361440958871019378336415511965329947939393439} a^{9} + \frac{3293794831905708622491375710220640767854991845}{16722881917742038756672831023930659895878786878} a^{8} - \frac{502568630127363705677883649773374740548924530}{8361440958871019378336415511965329947939393439} a^{7} - \frac{2084303479448006222162356581962621560423907770}{8361440958871019378336415511965329947939393439} a^{6} + \frac{3928988326429785316938946547951212168893017765}{8361440958871019378336415511965329947939393439} a^{5} + \frac{3638485985031995113927161813508136829951431439}{8361440958871019378336415511965329947939393439} a^{4} + \frac{492029195790107349682791268454287434442151180}{1194491565553002768333773644566475706848484777} a^{3} + \frac{1637442571663961797063453862907423498123690534}{8361440958871019378336415511965329947939393439} a^{2} - \frac{1779826831262150500635106351476099803116792495}{8361440958871019378336415511965329947939393439} a - \frac{1157983220550047615171453334985181805375992424}{8361440958871019378336415511965329947939393439}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{1446}$, which has order $11568$ (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:  \( 20726.065235 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1086:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 97 conjugacy class representatives for t16n1086 are not computed
Character table for t16n1086 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.4.7168.1, 4.4.14336.1, 8.8.3288334336.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
$73$73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.4.2.2$x^{4} - 73 x^{2} + 58619$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$