↑

Properties

Label 16.4.513...056.3
Degree $16$
Signature $[4, 6]$
Discriminant $5.140\times 10^{19}$
Root discriminant \(17.06\)
Ramified primes $2,3,7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^2\wr C_2$ (as 16T46)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 4*x^13 + 27*x^12 + 18*x^11 - 62*x^10 - 24*x^9 + 112*x^8 + 10*x^7 - 186*x^6 - 16*x^5 + 154*x^4 - 42*x^3 - 66*x^2 + 30*x - 3)
 
gp: K = bnfinit(y^16 - 8*y^14 - 4*y^13 + 27*y^12 + 18*y^11 - 62*y^10 - 24*y^9 + 112*y^8 + 10*y^7 - 186*y^6 - 16*y^5 + 154*y^4 - 42*y^3 - 66*y^2 + 30*y - 3, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 - 4*x^13 + 27*x^12 + 18*x^11 - 62*x^10 - 24*x^9 + 112*x^8 + 10*x^7 - 186*x^6 - 16*x^5 + 154*x^4 - 42*x^3 - 66*x^2 + 30*x - 3);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^14 - 4*x^13 + 27*x^12 + 18*x^11 - 62*x^10 - 24*x^9 + 112*x^8 + 10*x^7 - 186*x^6 - 16*x^5 + 154*x^4 - 42*x^3 - 66*x^2 + 30*x - 3)
 

\( x^{16} - 8 x^{14} - 4 x^{13} + 27 x^{12} + 18 x^{11} - 62 x^{10} - 24 x^{9} + 112 x^{8} + 10 x^{7} + \cdots - 3 \) 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:   \(51399544780206637056\) \(\medspace = 2^{24}\cdot 3^{12}\cdot 7^{8}\) 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:  \(17.06\)
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:  $2^{3/2}3^{3/4}7^{1/2}\approx 17.058268835716344$
Ramified primes:   \(2\), \(3\), \(7\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}+\frac{2}{7}a^{11}-\frac{3}{7}a^{10}-\frac{3}{7}a^{9}-\frac{1}{7}a^{8}+\frac{3}{7}a^{7}+\frac{1}{7}a^{6}+\frac{3}{7}a^{4}-\frac{3}{7}a^{3}-\frac{1}{7}a^{2}+\frac{1}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{13}+\frac{3}{7}a^{10}-\frac{2}{7}a^{9}-\frac{2}{7}a^{8}+\frac{2}{7}a^{7}-\frac{2}{7}a^{6}+\frac{3}{7}a^{5}-\frac{2}{7}a^{4}-\frac{2}{7}a^{3}+\frac{3}{7}a^{2}+\frac{2}{7}a-\frac{1}{7}$, $\frac{1}{7}a^{14}+\frac{3}{7}a^{11}-\frac{2}{7}a^{10}-\frac{2}{7}a^{9}+\frac{2}{7}a^{8}-\frac{2}{7}a^{7}+\frac{3}{7}a^{6}-\frac{2}{7}a^{5}-\frac{2}{7}a^{4}+\frac{3}{7}a^{3}+\frac{2}{7}a^{2}-\frac{1}{7}a$, $\frac{1}{7913077627}a^{15}+\frac{159674771}{7913077627}a^{14}+\frac{41762464}{1130439661}a^{13}-\frac{51811276}{7913077627}a^{12}-\frac{1667878080}{7913077627}a^{11}-\frac{1102380815}{7913077627}a^{10}-\frac{453888665}{1130439661}a^{9}-\frac{1338955078}{7913077627}a^{8}-\frac{3168314104}{7913077627}a^{7}-\frac{593590934}{7913077627}a^{6}-\frac{96341221}{608698279}a^{5}-\frac{193089705}{7913077627}a^{4}-\frac{3027520434}{7913077627}a^{3}-\frac{682235363}{7913077627}a^{2}-\frac{1070908733}{7913077627}a-\frac{1725374876}{7913077627}$ 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:   $\frac{307502610}{7913077627}a^{15}+\frac{680885781}{7913077627}a^{14}-\frac{3264650488}{7913077627}a^{13}-\frac{5929720423}{7913077627}a^{12}+\frac{10983590915}{7913077627}a^{11}+\frac{22100505222}{7913077627}a^{10}-\frac{24326179867}{7913077627}a^{9}-\frac{46335393631}{7913077627}a^{8}+\frac{57359670492}{7913077627}a^{7}+\frac{56578834313}{7913077627}a^{6}-\frac{8017743926}{608698279}a^{5}-\frac{83843833512}{7913077627}a^{4}+\frac{17012195063}{1130439661}a^{3}+\frac{30584524910}{7913077627}a^{2}-\frac{10851299476}{1130439661}a+\frac{22881288265}{7913077627}$, $\frac{181960035}{7913077627}a^{15}+\frac{144634983}{7913077627}a^{14}-\frac{1504613680}{7913077627}a^{13}-\frac{279041222}{1130439661}a^{12}+\frac{4837170111}{7913077627}a^{11}+\frac{7610658194}{7913077627}a^{10}-\frac{10254699162}{7913077627}a^{9}-\frac{14144050162}{7913077627}a^{8}+\frac{2837986128}{1130439661}a^{7}+\frac{17596846348}{7913077627}a^{6}-\frac{2902335117}{608698279}a^{5}-\frac{25944255276}{7913077627}a^{4}+\frac{4210407947}{1130439661}a^{3}+\frac{1772818838}{1130439661}a^{2}-\frac{8569163921}{7913077627}a-\frac{2504590544}{7913077627}$, $\frac{1497862295}{7913077627}a^{15}-\frac{63121008}{7913077627}a^{14}-\frac{11768181356}{7913077627}a^{13}-\frac{5607468781}{7913077627}a^{12}+\frac{39252420936}{7913077627}a^{11}+\frac{25135906463}{7913077627}a^{10}-\frac{89345384257}{7913077627}a^{9}-\frac{31642424869}{7913077627}a^{8}+\frac{158700815393}{7913077627}a^{7}+\frac{13407979968}{7913077627}a^{6}-\frac{20336554151}{608698279}a^{5}-\frac{26801915499}{7913077627}a^{4}+\frac{207670165929}{7913077627}a^{3}-\frac{46325438759}{7913077627}a^{2}-\frac{90457517526}{7913077627}a+\frac{2933503361}{1130439661}$, $\frac{2868150861}{7913077627}a^{15}-\frac{1573567648}{7913077627}a^{14}-\frac{21097535609}{7913077627}a^{13}-\frac{132168051}{1130439661}a^{12}+\frac{71059445673}{7913077627}a^{11}+\frac{15842712692}{7913077627}a^{10}-\frac{162461408231}{7913077627}a^{9}+\frac{13915310112}{7913077627}a^{8}+\frac{36559262242}{1130439661}a^{7}-\frac{80334648047}{7913077627}a^{6}-\frac{31154447147}{608698279}a^{5}+\frac{109617317941}{7913077627}a^{4}+\frac{256994894066}{7913077627}a^{3}-\frac{162328160374}{7913077627}a^{2}-\frac{30007199435}{7913077627}a+\frac{1607016886}{1130439661}$, $\frac{157503977}{7913077627}a^{15}+\frac{45507601}{1130439661}a^{14}-\frac{1170172987}{7913077627}a^{13}-\frac{450696594}{1130439661}a^{12}+\frac{2393017108}{7913077627}a^{11}+\frac{10763325617}{7913077627}a^{10}-\frac{2525201912}{7913077627}a^{9}-\frac{21105637348}{7913077627}a^{8}+\frac{983675725}{1130439661}a^{7}+\frac{34129274042}{7913077627}a^{6}-\frac{207306748}{86956897}a^{5}-\frac{58428769719}{7913077627}a^{4}+\frac{2931518322}{7913077627}a^{3}+\frac{34014113963}{7913077627}a^{2}-\frac{12809922150}{7913077627}a-\frac{16126150601}{7913077627}$, $\frac{2197829658}{7913077627}a^{15}-\frac{74189190}{1130439661}a^{14}-\frac{16675738654}{7913077627}a^{13}-\frac{5597603147}{7913077627}a^{12}+\frac{55072664526}{7913077627}a^{11}+\frac{28213419968}{7913077627}a^{10}-\frac{123302998082}{7913077627}a^{9}-\frac{24623888589}{7913077627}a^{8}+\frac{206682553104}{7913077627}a^{7}-\frac{10133064888}{7913077627}a^{6}-\frac{26165379742}{608698279}a^{5}+\frac{2745280128}{7913077627}a^{4}+\frac{230186936856}{7913077627}a^{3}-\frac{86614791775}{7913077627}a^{2}-\frac{9767967321}{1130439661}a+\frac{18749275085}{7913077627}$, $\frac{243010816}{7913077627}a^{15}+\frac{73444582}{7913077627}a^{14}-\frac{1740760144}{7913077627}a^{13}-\frac{1428181858}{7913077627}a^{12}+\frac{4517572753}{7913077627}a^{11}+\frac{4680422265}{7913077627}a^{10}-\frac{8091809695}{7913077627}a^{9}-\frac{559766757}{1130439661}a^{8}+\frac{13228943457}{7913077627}a^{7}-\frac{215856455}{7913077627}a^{6}-\frac{1695781768}{608698279}a^{5}-\frac{6104518577}{7913077627}a^{4}+\frac{1255165066}{7913077627}a^{3}-\frac{2412512655}{1130439661}a^{2}+\frac{264267034}{7913077627}a+\frac{6197273995}{7913077627}$, $\frac{2147274601}{7913077627}a^{15}+\frac{30134319}{1130439661}a^{14}-\frac{16697775244}{7913077627}a^{13}-\frac{10628967922}{7913077627}a^{12}+\frac{53552658648}{7913077627}a^{11}+\frac{44993581257}{7913077627}a^{10}-\frac{116812387775}{7913077627}a^{9}-\frac{64628280156}{7913077627}a^{8}+\frac{29467231179}{1130439661}a^{7}+\frac{52388615236}{7913077627}a^{6}-\frac{27178030046}{608698279}a^{5}-\frac{96425982613}{7913077627}a^{4}+\frac{264219063382}{7913077627}a^{3}-\frac{25619619379}{7913077627}a^{2}-\frac{16655083624}{1130439661}a+\frac{16136493890}{7913077627}$, $\frac{35506655}{7913077627}a^{15}-\frac{764945176}{7913077627}a^{14}+\frac{405349545}{7913077627}a^{13}+\frac{5271717756}{7913077627}a^{12}-\frac{87311983}{1130439661}a^{11}-\frac{18013127983}{7913077627}a^{10}-\frac{728859283}{7913077627}a^{9}+\frac{42542265468}{7913077627}a^{8}-\frac{11825263399}{7913077627}a^{7}-\frac{62680436353}{7913077627}a^{6}+\frac{2265886346}{608698279}a^{5}+\frac{95004253819}{7913077627}a^{4}-\frac{44001669510}{7913077627}a^{3}-\frac{7800509525}{1130439661}a^{2}+\frac{43522429396}{7913077627}a-\frac{1255316611}{1130439661}$ 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:  \( 5981.48582831 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 5981.48582831 \cdot 1}{2\cdot\sqrt{51399544780206637056}}\cr\approx \mathstrut & 0.410675352809 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 4*x^13 + 27*x^12 + 18*x^11 - 62*x^10 - 24*x^9 + 112*x^8 + 10*x^7 - 186*x^6 - 16*x^5 + 154*x^4 - 42*x^3 - 66*x^2 + 30*x - 3)
 
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^14 - 4*x^13 + 27*x^12 + 18*x^11 - 62*x^10 - 24*x^9 + 112*x^8 + 10*x^7 - 186*x^6 - 16*x^5 + 154*x^4 - 42*x^3 - 66*x^2 + 30*x - 3, 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^14 - 4*x^13 + 27*x^12 + 18*x^11 - 62*x^10 - 24*x^9 + 112*x^8 + 10*x^7 - 186*x^6 - 16*x^5 + 154*x^4 - 42*x^3 - 66*x^2 + 30*x - 3);
 
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^14 - 4*x^13 + 27*x^12 + 18*x^11 - 62*x^10 - 24*x^9 + 112*x^8 + 10*x^7 - 186*x^6 - 16*x^5 + 154*x^4 - 42*x^3 - 66*x^2 + 30*x - 3);
 
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

$C_2^2\wr C_2$ (as 16T46):

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 14 conjugacy class representatives for $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{3}) \), 4.2.21168.1, 4.2.1323.1, 4.2.9408.2, 4.2.9408.1, 4.2.84672.5, \(\Q(\sqrt{3}, \sqrt{7})\), 4.2.1728.1, 8.4.7169347584.1, 8.4.448084224.2, 8.4.796594176.1

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: 8.0.9144576.1, 8.0.9144576.2, 8.0.146313216.1, 8.0.7169347584.8, 8.0.146313216.2, 8.0.7169347584.4, 8.0.448084224.2, 8.0.448084224.1
Degree 16 siblings: 16.0.51399544780206637056.2, 16.0.21407557176262656.1, 16.0.200779471797682176.1, 16.0.51399544780206637056.8, 16.0.51399544780206637056.4, 16.0.51399544780206637056.7
Minimal sibling: 8.0.9144576.1

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ R ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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$$4$$4$$24$
\(3\) Copy content Toggle raw display 3.4.3.2$x^{4} + 6$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.2$x^{4} + 6$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
\(7\) Copy content Toggle raw display 7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$