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Properties

Label 16.4.15998750436...201.20
Degree $16$
Signature $[4, 6]$
Discriminant $41^{14}\cdot 59^{6}$
Root discriminant $118.92$
Ramified primes $41, 59$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![775024, -3270488, 5084664, -1701254, -5089343, 4422342, 1590774, -1842376, 75018, 287902, -71858, -5822, 3554, 0, -74, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 74*x^14 + 3554*x^12 - 5822*x^11 - 71858*x^10 + 287902*x^9 + 75018*x^8 - 1842376*x^7 + 1590774*x^6 + 4422342*x^5 - 5089343*x^4 - 1701254*x^3 + 5084664*x^2 - 3270488*x + 775024)
 
gp: K = bnfinit(x^16 - 74*x^14 + 3554*x^12 - 5822*x^11 - 71858*x^10 + 287902*x^9 + 75018*x^8 - 1842376*x^7 + 1590774*x^6 + 4422342*x^5 - 5089343*x^4 - 1701254*x^3 + 5084664*x^2 - 3270488*x + 775024, 1)
 

Normalized defining polynomial

\( x^{16} - 74 x^{14} + 3554 x^{12} - 5822 x^{11} - 71858 x^{10} + 287902 x^{9} + 75018 x^{8} - 1842376 x^{7} + 1590774 x^{6} + 4422342 x^{5} - 5089343 x^{4} - 1701254 x^{3} + 5084664 x^{2} - 3270488 x + 775024 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1599875043672267792208865723998201=41^{14}\cdot 59^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $118.92$
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:  $41, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} + \frac{1}{16} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{16} a^{5} + \frac{1}{16} a^{4} - \frac{1}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{12} + \frac{1}{16} a^{9} + \frac{3}{16} a^{6} - \frac{5}{16} a^{3} - \frac{1}{2}$, $\frac{1}{21712} a^{13} + \frac{503}{21712} a^{12} - \frac{275}{21712} a^{11} + \frac{134}{1357} a^{10} + \frac{419}{5428} a^{9} - \frac{37}{5428} a^{8} + \frac{2719}{21712} a^{7} + \frac{1777}{21712} a^{6} - \frac{1917}{21712} a^{5} - \frac{427}{5428} a^{4} + \frac{919}{2714} a^{3} + \frac{339}{2714} a^{2} - \frac{5}{23} a - \frac{7}{23}$, $\frac{1}{42946336} a^{14} - \frac{577}{42946336} a^{13} - \frac{11571}{42946336} a^{12} - \frac{26337}{998752} a^{11} - \frac{3221677}{42946336} a^{10} - \frac{1917431}{42946336} a^{9} + \frac{1145027}{42946336} a^{8} + \frac{8990573}{42946336} a^{7} - \frac{1307713}{42946336} a^{6} - \frac{6812913}{42946336} a^{5} + \frac{6613705}{42946336} a^{4} - \frac{18056597}{42946336} a^{3} - \frac{579519}{1342073} a^{2} + \frac{22367}{45494} a - \frac{22577}{90988}$, $\frac{1}{22852698119046011794228711419539343232} a^{15} + \frac{114278655243741383471446735855}{11426349059523005897114355709769671616} a^{14} - \frac{155033572032760618686476635171111}{11426349059523005897114355709769671616} a^{13} - \frac{46475345188708935548611233500607991}{5713174529761502948557177854884835808} a^{12} - \frac{354608905938588476257010931955741199}{11426349059523005897114355709769671616} a^{11} - \frac{1196888825495157813101474001849066693}{11426349059523005897114355709769671616} a^{10} - \frac{1256983030422353787103825620669457659}{11426349059523005897114355709769671616} a^{9} - \frac{914011587714879970441081585699681527}{11426349059523005897114355709769671616} a^{8} + \frac{2260388082713649870213082878372044007}{11426349059523005897114355709769671616} a^{7} - \frac{535114563230734264296371000142071315}{5713174529761502948557177854884835808} a^{6} - \frac{1411603019413090362149087968235276173}{11426349059523005897114355709769671616} a^{5} - \frac{1362887479551665114606710334141455055}{11426349059523005897114355709769671616} a^{4} + \frac{1328516590983602621109605316073457013}{22852698119046011794228711419539343232} a^{3} - \frac{540318117610272180101463406765071967}{2856587264880751474278588927442417904} a^{2} + \frac{19990294749085554549325926279850287}{48416733303063584309806591990549456} a - \frac{14264877859027500055672899802231343}{48416733303063584309806591990549456}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 88858553706.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 4.2.99179.1, 4.2.4066339.1, 8.4.677939627379761.3 x2, 8.4.16535112862921.1

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 8 siblings: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ R

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
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
$59$$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$