Properties

Label 16.8.123...257.1
Degree $16$
Signature $[8, 4]$
Discriminant $1.233\times 10^{38}$
Root discriminant $240.26$
Ramified primes $7, 73$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 29*x^14 + 630*x^13 - 4609*x^12 - 33136*x^11 + 168901*x^10 + 1035038*x^9 - 2476072*x^8 - 17969440*x^7 + 16115584*x^6 + 185656320*x^5 + 24797184*x^4 - 1016004608*x^3 - 749731840*x^2 + 2181038080*x + 2147483648)
 
gp: K = bnfinit(x^16 - 4*x^15 - 29*x^14 + 630*x^13 - 4609*x^12 - 33136*x^11 + 168901*x^10 + 1035038*x^9 - 2476072*x^8 - 17969440*x^7 + 16115584*x^6 + 185656320*x^5 + 24797184*x^4 - 1016004608*x^3 - 749731840*x^2 + 2181038080*x + 2147483648, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2147483648, 2181038080, -749731840, -1016004608, 24797184, 185656320, 16115584, -17969440, -2476072, 1035038, 168901, -33136, -4609, 630, -29, -4, 1]);
 

\( x^{16} - 4 x^{15} - 29 x^{14} + 630 x^{13} - 4609 x^{12} - 33136 x^{11} + 168901 x^{10} + 1035038 x^{9} - 2476072 x^{8} - 17969440 x^{7} + 16115584 x^{6} + 185656320 x^{5} + 24797184 x^{4} - 1016004608 x^{3} - 749731840 x^{2} + 2181038080 x + 2147483648 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(123315986626517312705762123587383695257\)\(\medspace = 7^{12}\cdot 73^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $240.26$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 73$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{64} a^{7} - \frac{1}{32} a^{6} - \frac{1}{32} a^{5} - \frac{1}{16} a^{4} + \frac{9}{64} a^{3} + \frac{3}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{128} a^{8} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} - \frac{7}{128} a^{4} + \frac{3}{16} a^{3} + \frac{1}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{256} a^{9} - \frac{1}{256} a^{8} - \frac{3}{128} a^{6} - \frac{11}{256} a^{5} + \frac{7}{256} a^{4} + \frac{21}{128} a^{3} - \frac{1}{8} a$, $\frac{1}{2048} a^{10} + \frac{1}{2048} a^{9} + \frac{1}{512} a^{8} + \frac{3}{1024} a^{7} - \frac{19}{2048} a^{6} + \frac{89}{2048} a^{5} - \frac{41}{1024} a^{4} + \frac{9}{64} a^{3} - \frac{9}{64} a^{2}$, $\frac{1}{4096} a^{11} + \frac{3}{4096} a^{9} + \frac{1}{2048} a^{8} + \frac{7}{4096} a^{7} + \frac{11}{1024} a^{6} - \frac{235}{4096} a^{5} - \frac{7}{2048} a^{4} - \frac{25}{128} a^{3} + \frac{23}{128} a^{2} + \frac{1}{16} a$, $\frac{1}{32768} a^{12} + \frac{3}{32768} a^{10} + \frac{17}{16384} a^{9} - \frac{121}{32768} a^{8} + \frac{27}{8192} a^{7} - \frac{1003}{32768} a^{6} - \frac{503}{16384} a^{5} + \frac{51}{1024} a^{4} + \frac{51}{1024} a^{3} + \frac{27}{128} a^{2} - \frac{1}{4} a$, $\frac{1}{262144} a^{13} - \frac{29}{262144} a^{11} + \frac{1}{131072} a^{10} - \frac{249}{262144} a^{9} - \frac{149}{65536} a^{8} - \frac{1419}{262144} a^{7} - \frac{391}{131072} a^{6} - \frac{123}{8192} a^{5} - \frac{445}{8192} a^{4} - \frac{157}{1024} a^{3} - \frac{1}{64} a^{2} + \frac{1}{4} a$, $\frac{1}{4194304} a^{14} + \frac{1}{1048576} a^{13} - \frac{61}{4194304} a^{12} - \frac{185}{2097152} a^{11} + \frac{431}{4194304} a^{10} + \frac{945}{524288} a^{9} - \frac{9659}{4194304} a^{8} - \frac{9693}{2097152} a^{7} - \frac{11367}{524288} a^{6} + \frac{309}{131072} a^{5} - \frac{447}{32768} a^{4} + \frac{33}{1024} a^{3} + \frac{19}{512} a^{2} - \frac{3}{32} a$, $\frac{1}{1994677812449869495550127212855296} a^{15} - \frac{42496337087057743890966977}{498669453112467373887531803213824} a^{14} + \frac{2363315238261840903532374371}{1994677812449869495550127212855296} a^{13} - \frac{12759013590047422521939125061}{997338906224934747775063606427648} a^{12} - \frac{229950576323245583492492825473}{1994677812449869495550127212855296} a^{11} - \frac{25131204558178493068799889847}{124667363278116843471882950803456} a^{10} + \frac{2539066406856120680951547484741}{1994677812449869495550127212855296} a^{9} + \frac{2871428291227508796775311259407}{997338906224934747775063606427648} a^{8} + \frac{1797536673267780077014732667947}{249334726556233686943765901606912} a^{7} + \frac{135122418598570013474024275247}{62333681639058421735941475401728} a^{6} - \frac{842980276234761668492546223841}{15583420409764605433985368850432} a^{5} - \frac{6423654936990939646434798147}{1947927551220575679248171106304} a^{4} - \frac{53251417137602872058756920069}{243490943902571959906021388288} a^{3} - \frac{6424904042918485690189547091}{30436367987821494988252673536} a^{2} + \frac{650110603336955734967254577}{1902272999238843436765792096} a + \frac{27202703070299716210871110}{59446031226213857398931003}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 514588263316000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 514588263316000000 \cdot 1}{2\sqrt{123315986626517312705762123587383695257}}\approx 9244.41915394571$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.8.26524803844351897.1

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ R $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ $16$ $16$ $16$ $16$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
73Data not computed