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Properties

Label 16.4.891...000.1
Degree $16$
Signature $[4, 6]$
Discriminant $8.916\times 10^{20}$
Root discriminant \(20.39\)
Ramified primes $2,3,5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_8:C_2$ (as 16T45)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 60*x^13 + 284*x^11 - 758*x^10 + 1276*x^9 - 1709*x^8 + 1428*x^7 - 246*x^6 - 452*x^5 + 30*x^4 + 420*x^3 - 326*x^2 + 100*x - 11)
 
gp: K = bnfinit(y^16 - 8*y^15 + 32*y^14 - 60*y^13 + 284*y^11 - 758*y^10 + 1276*y^9 - 1709*y^8 + 1428*y^7 - 246*y^6 - 452*y^5 + 30*y^4 + 420*y^3 - 326*y^2 + 100*y - 11, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 32*x^14 - 60*x^13 + 284*x^11 - 758*x^10 + 1276*x^9 - 1709*x^8 + 1428*x^7 - 246*x^6 - 452*x^5 + 30*x^4 + 420*x^3 - 326*x^2 + 100*x - 11);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 32*x^14 - 60*x^13 + 284*x^11 - 758*x^10 + 1276*x^9 - 1709*x^8 + 1428*x^7 - 246*x^6 - 452*x^5 + 30*x^4 + 420*x^3 - 326*x^2 + 100*x - 11)
 

\( x^{16} - 8 x^{15} + 32 x^{14} - 60 x^{13} + 284 x^{11} - 758 x^{10} + 1276 x^{9} - 1709 x^{8} + \cdots - 11 \) Copy content Toggle raw display

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[4, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(891610044825600000000\) \(\medspace = 2^{32}\cdot 3^{12}\cdot 5^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(20.39\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{2}3^{3/4}5^{1/2}\approx 20.38853093816547$
Ramified primes:   \(2\), \(3\), \(5\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{935}a^{14}+\frac{1}{11}a^{13}-\frac{79}{935}a^{12}+\frac{328}{935}a^{11}-\frac{82}{935}a^{10}+\frac{2}{17}a^{9}+\frac{129}{935}a^{8}+\frac{128}{935}a^{7}-\frac{4}{85}a^{6}-\frac{212}{935}a^{5}-\frac{118}{935}a^{4}+\frac{291}{935}a^{3}-\frac{354}{935}a^{2}+\frac{392}{935}a-\frac{36}{85}$, $\frac{1}{390632077536635}a^{15}-\frac{121465511742}{390632077536635}a^{14}+\frac{3147051251147}{7370416557295}a^{13}-\frac{2542978515849}{390632077536635}a^{12}-\frac{189237969558808}{390632077536635}a^{11}-\frac{13567993235961}{35512007048785}a^{10}+\frac{152338105896424}{390632077536635}a^{9}-\frac{4617357513079}{78126415507327}a^{8}-\frac{3296219276226}{7102401409757}a^{7}+\frac{111153633295141}{390632077536635}a^{6}+\frac{134089093000756}{390632077536635}a^{5}-\frac{108624331215043}{390632077536635}a^{4}+\frac{98834669461834}{390632077536635}a^{3}-\frac{13753454861358}{78126415507327}a^{2}+\frac{380258631091}{7102401409757}a-\frac{10353526401448}{35512007048785}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{741161916582}{4595671500431}a^{15}-\frac{471904075122411}{390632077536635}a^{14}+\frac{387724083718}{86710783027}a^{13}-\frac{27\!\cdots\!16}{390632077536635}a^{12}-\frac{20\!\cdots\!88}{390632077536635}a^{11}+\frac{17\!\cdots\!47}{390632077536635}a^{10}-\frac{74\!\cdots\!48}{78126415507327}a^{9}+\frac{54\!\cdots\!41}{390632077536635}a^{8}-\frac{68\!\cdots\!28}{390632077536635}a^{7}+\frac{38\!\cdots\!19}{390632077536635}a^{6}+\frac{22\!\cdots\!82}{390632077536635}a^{5}-\frac{21\!\cdots\!12}{390632077536635}a^{4}-\frac{18\!\cdots\!46}{390632077536635}a^{3}+\frac{19\!\cdots\!64}{390632077536635}a^{2}-\frac{47\!\cdots\!02}{390632077536635}a-\frac{1788606912414}{35512007048785}$, $\frac{478796504857252}{390632077536635}a^{15}-\frac{34\!\cdots\!23}{390632077536635}a^{14}+\frac{241201221513299}{7370416557295}a^{13}-\frac{19\!\cdots\!27}{390632077536635}a^{12}-\frac{14\!\cdots\!48}{390632077536635}a^{11}+\frac{12\!\cdots\!91}{390632077536635}a^{10}-\frac{27\!\cdots\!12}{390632077536635}a^{9}+\frac{41\!\cdots\!39}{390632077536635}a^{8}-\frac{51\!\cdots\!52}{390632077536635}a^{7}+\frac{30\!\cdots\!58}{390632077536635}a^{6}+\frac{19\!\cdots\!22}{7102401409757}a^{5}-\frac{14\!\cdots\!54}{390632077536635}a^{4}-\frac{88\!\cdots\!46}{390632077536635}a^{3}+\frac{13\!\cdots\!61}{390632077536635}a^{2}-\frac{55\!\cdots\!43}{390632077536635}a+\frac{653371163600513}{35512007048785}$, $\frac{113893646030138}{78126415507327}a^{15}-\frac{41\!\cdots\!86}{390632077536635}a^{14}+\frac{57359198674961}{1474083311459}a^{13}-\frac{23\!\cdots\!11}{390632077536635}a^{12}-\frac{17\!\cdots\!88}{390632077536635}a^{11}+\frac{13\!\cdots\!32}{35512007048785}a^{10}-\frac{64\!\cdots\!36}{78126415507327}a^{9}+\frac{48\!\cdots\!51}{390632077536635}a^{8}-\frac{55\!\cdots\!48}{35512007048785}a^{7}+\frac{35\!\cdots\!39}{390632077536635}a^{6}+\frac{13\!\cdots\!12}{390632077536635}a^{5}-\frac{16\!\cdots\!12}{390632077536635}a^{4}-\frac{11\!\cdots\!76}{390632077536635}a^{3}+\frac{16\!\cdots\!59}{390632077536635}a^{2}-\frac{56\!\cdots\!27}{35512007048785}a+\frac{731424369731521}{35512007048785}$, $\frac{583173079081952}{390632077536635}a^{15}-\frac{854887401959563}{78126415507327}a^{14}+\frac{298231776472939}{7370416557295}a^{13}-\frac{24\!\cdots\!59}{390632077536635}a^{12}-\frac{947205306312022}{22978357502155}a^{11}+\frac{30\!\cdots\!05}{78126415507327}a^{10}-\frac{33\!\cdots\!57}{390632077536635}a^{9}+\frac{52\!\cdots\!61}{390632077536635}a^{8}-\frac{65\!\cdots\!08}{390632077536635}a^{7}+\frac{40\!\cdots\!26}{390632077536635}a^{6}+\frac{10\!\cdots\!39}{35512007048785}a^{5}-\frac{17\!\cdots\!93}{390632077536635}a^{4}-\frac{98\!\cdots\!88}{390632077536635}a^{3}+\frac{17\!\cdots\!74}{390632077536635}a^{2}-\frac{75\!\cdots\!12}{390632077536635}a+\frac{199320536750792}{7102401409757}$, $\frac{178564821863394}{390632077536635}a^{15}-\frac{246226809283289}{78126415507327}a^{14}+\frac{81692534862853}{7370416557295}a^{13}-\frac{57\!\cdots\!68}{390632077536635}a^{12}-\frac{68\!\cdots\!38}{390632077536635}a^{11}+\frac{87\!\cdots\!08}{78126415507327}a^{10}-\frac{85\!\cdots\!79}{390632077536635}a^{9}+\frac{12\!\cdots\!02}{390632077536635}a^{8}-\frac{15\!\cdots\!16}{390632077536635}a^{7}+\frac{71\!\cdots\!82}{390632077536635}a^{6}+\frac{47\!\cdots\!33}{390632077536635}a^{5}-\frac{27\!\cdots\!41}{390632077536635}a^{4}-\frac{37\!\cdots\!91}{390632077536635}a^{3}+\frac{33\!\cdots\!93}{35512007048785}a^{2}-\frac{12\!\cdots\!49}{390632077536635}a+\frac{27128459038704}{7102401409757}$, $\frac{680768861845986}{390632077536635}a^{15}-\frac{48\!\cdots\!74}{390632077536635}a^{14}+\frac{326895383473097}{7370416557295}a^{13}-\frac{24\!\cdots\!26}{390632077536635}a^{12}-\frac{23\!\cdots\!84}{390632077536635}a^{11}+\frac{17\!\cdots\!63}{390632077536635}a^{10}-\frac{35\!\cdots\!91}{390632077536635}a^{9}+\frac{48\!\cdots\!02}{35512007048785}a^{8}-\frac{65\!\cdots\!56}{390632077536635}a^{7}+\frac{34\!\cdots\!59}{390632077536635}a^{6}+\frac{35\!\cdots\!04}{78126415507327}a^{5}-\frac{15\!\cdots\!72}{390632077536635}a^{4}-\frac{13\!\cdots\!48}{390632077536635}a^{3}+\frac{16\!\cdots\!68}{390632077536635}a^{2}-\frac{59\!\cdots\!04}{390632077536635}a+\frac{671828588617279}{35512007048785}$, $\frac{69778860653410}{78126415507327}a^{15}-\frac{26\!\cdots\!72}{390632077536635}a^{14}+\frac{38177748304843}{1474083311459}a^{13}-\frac{17\!\cdots\!37}{390632077536635}a^{12}-\frac{61\!\cdots\!21}{390632077536635}a^{11}+\frac{95\!\cdots\!34}{390632077536635}a^{10}-\frac{45\!\cdots\!70}{78126415507327}a^{9}+\frac{32\!\cdots\!37}{35512007048785}a^{8}-\frac{46\!\cdots\!81}{390632077536635}a^{7}+\frac{33\!\cdots\!08}{390632077536635}a^{6}+\frac{22\!\cdots\!34}{390632077536635}a^{5}-\frac{13\!\cdots\!39}{390632077536635}a^{4}-\frac{34\!\cdots\!32}{390632077536635}a^{3}+\frac{12\!\cdots\!33}{390632077536635}a^{2}-\frac{69\!\cdots\!74}{390632077536635}a+\frac{11\!\cdots\!87}{35512007048785}$, $\frac{2968487375668}{22978357502155}a^{15}-\frac{59758992172061}{78126415507327}a^{14}+\frac{963841204841}{433553915135}a^{13}-\frac{346010566296702}{390632077536635}a^{12}-\frac{38\!\cdots\!82}{390632077536635}a^{11}+\frac{21\!\cdots\!91}{78126415507327}a^{10}-\frac{11\!\cdots\!76}{390632077536635}a^{9}+\frac{92\!\cdots\!63}{390632077536635}a^{8}-\frac{49\!\cdots\!54}{390632077536635}a^{7}-\frac{26\!\cdots\!77}{390632077536635}a^{6}+\frac{36\!\cdots\!32}{390632077536635}a^{5}+\frac{11\!\cdots\!06}{390632077536635}a^{4}-\frac{26\!\cdots\!69}{390632077536635}a^{3}-\frac{101697515205003}{35512007048785}a^{2}+\frac{99\!\cdots\!19}{390632077536635}a-\frac{62730269515360}{7102401409757}$, $\frac{19399070228347}{78126415507327}a^{15}-\frac{761234322963597}{390632077536635}a^{14}+\frac{11174263393565}{1474083311459}a^{13}-\frac{52\!\cdots\!12}{390632077536635}a^{12}-\frac{13\!\cdots\!61}{390632077536635}a^{11}+\frac{27\!\cdots\!39}{390632077536635}a^{10}-\frac{13\!\cdots\!91}{78126415507327}a^{9}+\frac{10\!\cdots\!87}{390632077536635}a^{8}-\frac{14\!\cdots\!21}{390632077536635}a^{7}+\frac{10\!\cdots\!08}{390632077536635}a^{6}+\frac{56\!\cdots\!74}{390632077536635}a^{5}-\frac{44\!\cdots\!44}{390632077536635}a^{4}-\frac{527526262670796}{22978357502155}a^{3}+\frac{35\!\cdots\!13}{35512007048785}a^{2}-\frac{20\!\cdots\!99}{390632077536635}a+\frac{335100850094597}{35512007048785}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 20026.6033091 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 20026.6033091 \cdot 1}{2\cdot\sqrt{891610044825600000000}}\cr\approx \mathstrut & 0.330133059077 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 60*x^13 + 284*x^11 - 758*x^10 + 1276*x^9 - 1709*x^8 + 1428*x^7 - 246*x^6 - 452*x^5 + 30*x^4 + 420*x^3 - 326*x^2 + 100*x - 11)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 60*x^13 + 284*x^11 - 758*x^10 + 1276*x^9 - 1709*x^8 + 1428*x^7 - 246*x^6 - 452*x^5 + 30*x^4 + 420*x^3 - 326*x^2 + 100*x - 11, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 8*x^15 + 32*x^14 - 60*x^13 + 284*x^11 - 758*x^10 + 1276*x^9 - 1709*x^8 + 1428*x^7 - 246*x^6 - 452*x^5 + 30*x^4 + 420*x^3 - 326*x^2 + 100*x - 11);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 32*x^14 - 60*x^13 + 284*x^11 - 758*x^10 + 1276*x^9 - 1709*x^8 + 1428*x^7 - 246*x^6 - 452*x^5 + 30*x^4 + 420*x^3 - 326*x^2 + 100*x - 11);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$D_8:C_2$ (as 16T45):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{15}) \), 4.2.400.1, 4.2.3600.1, \(\Q(\sqrt{3}, \sqrt{5})\), 8.4.207360000.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 32
Degree 8 siblings: 8.0.5971968000.1, 8.0.5971968000.2
Degree 16 siblings: 16.0.35664401793024000000.1, 16.0.891610044825600000000.12, 16.0.891610044825600000000.9
Minimal sibling: 8.0.5971968000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $16$$8$$2$$32$
\(3\) Copy content Toggle raw display 3.16.12.2$x^{16} + 12 x^{12} + 36 x^{8} + 324$$4$$4$$12$$C_8: C_2$$[\ ]_{4}^{4}$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$