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Properties

Label 16.0.157...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.577\times 10^{23}$
Root discriminant \(28.17\)
Ramified primes $5,71$
Class number $7$ (GRH)
Class group [7] (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^14 - 11*x^12 + 93*x^10 - 59*x^8 + 1860*x^6 - 4400*x^4 - 24000*x^2 + 160000)
 
gp: K = bnfinit(y^16 - 3*y^14 - 11*y^12 + 93*y^10 - 59*y^8 + 1860*y^6 - 4400*y^4 - 24000*y^2 + 160000, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^14 - 11*x^12 + 93*x^10 - 59*x^8 + 1860*x^6 - 4400*x^4 - 24000*x^2 + 160000);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^14 - 11*x^12 + 93*x^10 - 59*x^8 + 1860*x^6 - 4400*x^4 - 24000*x^2 + 160000)
 

\( x^{16} - 3x^{14} - 11x^{12} + 93x^{10} - 59x^{8} + 1860x^{6} - 4400x^{4} - 24000x^{2} + 160000 \) 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:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(157654670714297119140625\) \(\medspace = 5^{12}\cdot 71^{8}\) 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:  \(28.17\)
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:  $5^{3/4}71^{1/2}\approx 28.17452984545484$
Ramified primes:   \(5\), \(71\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{4}$, $\frac{1}{1180}a^{10}+\frac{3}{20}a^{8}+\frac{1}{20}a^{6}+\frac{7}{20}a^{4}-\frac{1}{20}a^{2}-\frac{1}{2}a-\frac{25}{59}$, $\frac{1}{2360}a^{11}+\frac{3}{40}a^{9}-\frac{9}{40}a^{7}+\frac{7}{40}a^{5}-\frac{1}{2}a^{4}-\frac{1}{40}a^{3}-\frac{25}{118}a$, $\frac{1}{47200}a^{12}-\frac{1}{4720}a^{11}-\frac{3}{47200}a^{10}+\frac{17}{80}a^{9}-\frac{129}{800}a^{8}+\frac{9}{80}a^{7}-\frac{73}{800}a^{6}-\frac{7}{80}a^{5}+\frac{399}{800}a^{4}-\frac{39}{80}a^{3}-\frac{1087}{2360}a^{2}+\frac{25}{236}a-\frac{11}{118}$, $\frac{1}{472000}a^{13}+\frac{17}{472000}a^{11}-\frac{869}{8000}a^{9}-\frac{653}{8000}a^{7}-\frac{661}{8000}a^{5}+\frac{2377}{11800}a^{3}+\frac{277}{590}a-\frac{1}{2}$, $\frac{1}{944000}a^{14}-\frac{3}{944000}a^{12}-\frac{1}{4720}a^{11}-\frac{11}{944000}a^{10}-\frac{3}{80}a^{9}+\frac{3527}{16000}a^{8}-\frac{11}{80}a^{7}-\frac{1}{16000}a^{6}+\frac{13}{80}a^{5}+\frac{93}{47200}a^{4}+\frac{21}{80}a^{3}-\frac{11}{2360}a^{2}-\frac{17}{118}a-\frac{3}{118}$, $\frac{1}{9440000}a^{15}-\frac{3}{9440000}a^{13}+\frac{389}{9440000}a^{11}-\frac{1}{2360}a^{10}-\frac{3273}{160000}a^{9}-\frac{3}{40}a^{8}+\frac{12399}{160000}a^{7}+\frac{9}{40}a^{6}+\frac{55553}{472000}a^{5}+\frac{13}{40}a^{4}-\frac{951}{2360}a^{3}-\frac{19}{40}a^{2}-\frac{66}{295}a+\frac{25}{118}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{7}$, which has order $7$ (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:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{93}{472000} a^{14} + \frac{279}{472000} a^{12} + \frac{1023}{472000} a^{10} - \frac{11}{8000} a^{8} + \frac{93}{8000} a^{6} - \frac{8649}{23600} a^{4} + \frac{1023}{1180} a^{2} + \frac{279}{59} \)  (order $10$) 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:   $\frac{93}{472000}a^{14}+\frac{121}{472000}a^{12}-\frac{1023}{472000}a^{10}+\frac{11}{8000}a^{8}-\frac{93}{8000}a^{6}+\frac{8649}{23600}a^{4}+\frac{33}{59}a^{2}-\frac{220}{59}$, $\frac{431}{9440000}a^{15}+\frac{3}{47200}a^{14}-\frac{5013}{9440000}a^{13}-\frac{1}{1180}a^{12}-\frac{9581}{9440000}a^{11}+\frac{17}{160000}a^{9}-\frac{871}{160000}a^{7}+\frac{51057}{472000}a^{5}+\frac{6229}{47200}a^{4}-\frac{22039}{23600}a^{3}-\frac{1683}{1180}a^{2}-\frac{2613}{1180}a-1$, $\frac{131}{4720000}a^{15}-\frac{33}{472000}a^{14}+\frac{3}{80000}a^{13}+\frac{279}{472000}a^{12}+\frac{8649}{4720000}a^{11}+\frac{1023}{472000}a^{10}-\frac{93}{80000}a^{9}-\frac{11}{8000}a^{8}+\frac{59}{80000}a^{7}+\frac{93}{8000}a^{6}+\frac{36343}{472000}a^{5}-\frac{121}{1180}a^{4}+\frac{11}{200}a^{3}+\frac{1023}{1180}a^{2}+\frac{186}{59}a+\frac{499}{118}$, $\frac{351}{4720000}a^{15}+\frac{213}{944000}a^{14}-\frac{1223}{4720000}a^{13}-\frac{279}{944000}a^{12}+\frac{11}{80000}a^{11}-\frac{1023}{944000}a^{10}-\frac{93}{80000}a^{9}+\frac{11}{16000}a^{8}+\frac{59}{80000}a^{7}-\frac{93}{16000}a^{6}+\frac{66289}{472000}a^{5}+\frac{21107}{47200}a^{4}-\frac{9303}{23600}a^{3}-\frac{1023}{2360}a^{2}+\frac{3}{10}a-\frac{110}{59}$, $\frac{507}{9440000}a^{15}+\frac{3}{47200}a^{14}+\frac{4659}{9440000}a^{13}+\frac{1}{1180}a^{12}-\frac{7717}{9440000}a^{11}+\frac{169}{160000}a^{9}+\frac{753}{160000}a^{7}+\frac{1859}{23600}a^{5}+\frac{6229}{47200}a^{4}+\frac{20741}{23600}a^{3}+\frac{1683}{1180}a^{2}-\frac{1107}{1180}a-\frac{1}{2}$, $\frac{279}{9440000}a^{15}+\frac{93}{472000}a^{14}-\frac{997}{9440000}a^{13}+\frac{191}{472000}a^{12}+\frac{3811}{9440000}a^{11}-\frac{2233}{472000}a^{10}-\frac{127}{160000}a^{9}-\frac{19}{8000}a^{8}+\frac{1801}{160000}a^{7}-\frac{203}{8000}a^{6}+\frac{16979}{472000}a^{5}+\frac{4557}{9440}a^{4}-\frac{9477}{23600}a^{3}+\frac{1133}{1180}a^{2}+\frac{2037}{1180}a-\frac{609}{59}$, $\frac{17}{9440000}a^{15}-\frac{147}{944000}a^{14}+\frac{269}{9440000}a^{13}+\frac{921}{944000}a^{12}-\frac{9947}{9440000}a^{11}-\frac{1023}{944000}a^{10}+\frac{279}{160000}a^{9}+\frac{11}{16000}a^{8}-\frac{177}{160000}a^{7}-\frac{93}{16000}a^{6}-\frac{15883}{472000}a^{5}-\frac{16267}{47200}a^{4}+\frac{6139}{23600}a^{3}+\frac{2603}{1180}a^{2}-\frac{1107}{590}a-\frac{169}{59}$ 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:  \( 46337.4951728 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 46337.4951728 \cdot 7}{10\cdot\sqrt{157654670714297119140625}}\cr\approx \mathstrut & 0.198434015021 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^14 - 11*x^12 + 93*x^10 - 59*x^8 + 1860*x^6 - 4400*x^4 - 24000*x^2 + 160000)
 
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 - 3*x^14 - 11*x^12 + 93*x^10 - 59*x^8 + 1860*x^6 - 4400*x^4 - 24000*x^2 + 160000, 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 - 3*x^14 - 11*x^12 + 93*x^10 - 59*x^8 + 1860*x^6 - 4400*x^4 - 24000*x^2 + 160000);
 
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 - 3*x^14 - 11*x^12 + 93*x^10 - 59*x^8 + 1860*x^6 - 4400*x^4 - 24000*x^2 + 160000);
 
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

$C_2^2:C_4$ (as 16T10):

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 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-71}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-355}) \), \(\Q(\sqrt{5}, \sqrt{-71})\), 4.2.8875.1 x2, 4.0.630125.1 x2, 4.0.25205.1 x2, 4.2.1775.1 x2, 4.4.630125.1, \(\Q(\zeta_{5})\), 8.0.397057515625.1, 8.0.15882300625.1, 8.0.397057515625.2, 8.4.397057515625.1 x2, 8.0.78765625.1 x2

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

Degree 8 siblings: 8.4.397057515625.1, 8.0.78765625.1
Minimal sibling: 8.0.78765625.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(5\) Copy content Toggle raw display 5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(71\) Copy content Toggle raw display 71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$