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Properties

Label 16.0.328...009.1
Degree $16$
Signature $[0, 8]$
Discriminant $3.286\times 10^{37}$
Root discriminant \(221.20\)
Ramified primes $47,53$
Class number $30$ (GRH)
Class group [30] (GRH)
Galois group $\OD_{16}:C_2$ (as 16T36)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 424*x^12 + 130963*x^10 + 1892100*x^8 + 7036545*x^6 + 5578674*x^4 + 1806187*x^2 + 339889)
 
gp: K = bnfinit(y^16 + 424*y^12 + 130963*y^10 + 1892100*y^8 + 7036545*y^6 + 5578674*y^4 + 1806187*y^2 + 339889, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 424*x^12 + 130963*x^10 + 1892100*x^8 + 7036545*x^6 + 5578674*x^4 + 1806187*x^2 + 339889);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 424*x^12 + 130963*x^10 + 1892100*x^8 + 7036545*x^6 + 5578674*x^4 + 1806187*x^2 + 339889)
 

\( x^{16} + 424x^{12} + 130963x^{10} + 1892100x^{8} + 7036545x^{6} + 5578674x^{4} + 1806187x^{2} + 339889 \) 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:   \(32858296023668216937367870328998288009\) \(\medspace = 47^{8}\cdot 53^{14}\) 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:  \(221.20\)
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:  $47^{1/2}53^{7/8}\approx 221.20264458677704$
Ramified primes:   \(47\), \(53\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{106}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{106}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{106}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{106}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{106}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{1166}a^{13}-\frac{5}{1166}a^{11}-\frac{1}{583}a^{9}-\frac{9}{22}a^{7}+\frac{3}{11}a^{5}-\frac{1}{2}a^{4}-\frac{1}{11}a^{3}-\frac{1}{2}a^{2}-\frac{1}{11}a$, $\frac{1}{39\!\cdots\!62}a^{14}+\frac{70\!\cdots\!25}{19\!\cdots\!81}a^{12}+\frac{13\!\cdots\!81}{39\!\cdots\!62}a^{10}-\frac{54\!\cdots\!33}{13\!\cdots\!78}a^{8}+\frac{56\!\cdots\!99}{73\!\cdots\!54}a^{6}-\frac{1}{2}a^{5}-\frac{11\!\cdots\!71}{36\!\cdots\!77}a^{4}-\frac{11\!\cdots\!56}{36\!\cdots\!77}a^{2}-\frac{1}{2}a-\frac{81\!\cdots\!32}{30\!\cdots\!37}$, $\frac{1}{20\!\cdots\!86}a^{15}-\frac{17\!\cdots\!83}{39\!\cdots\!62}a^{13}-\frac{50\!\cdots\!60}{19\!\cdots\!81}a^{11}-\frac{26\!\cdots\!59}{13\!\cdots\!78}a^{9}-\frac{55\!\cdots\!16}{19\!\cdots\!81}a^{7}-\frac{1}{2}a^{6}-\frac{43\!\cdots\!42}{36\!\cdots\!77}a^{5}-\frac{1}{2}a^{4}-\frac{60\!\cdots\!00}{36\!\cdots\!77}a^{3}-\frac{92\!\cdots\!66}{33\!\cdots\!07}a$ 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

$C_{30}$, which has order $30$ (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:   \( -1 \)  (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:   $\frac{82431245079}{11\!\cdots\!27}a^{14}-\frac{47928349647}{11\!\cdots\!27}a^{12}+\frac{659971768595}{222821459063159}a^{10}+\frac{203303992440904}{222821459063159}a^{8}+\frac{28\!\cdots\!25}{222821459063159}a^{6}+\frac{93\!\cdots\!86}{222821459063159}a^{4}+\frac{32\!\cdots\!09}{222821459063159}a^{2}-\frac{5635747641564}{1841499661679}$, $\frac{45\!\cdots\!89}{35\!\cdots\!17}a^{15}+\frac{19\!\cdots\!50}{67\!\cdots\!89}a^{13}+\frac{35\!\cdots\!29}{67\!\cdots\!89}a^{11}+\frac{11\!\cdots\!99}{67\!\cdots\!89}a^{9}+\frac{16\!\cdots\!86}{67\!\cdots\!89}a^{7}+\frac{11\!\cdots\!68}{12\!\cdots\!13}a^{5}+\frac{39\!\cdots\!06}{12\!\cdots\!13}a^{3}+\frac{10\!\cdots\!06}{11\!\cdots\!83}a$, $\frac{48\!\cdots\!98}{35\!\cdots\!17}a^{15}-\frac{430295401778089}{67\!\cdots\!89}a^{13}+\frac{40\!\cdots\!12}{67\!\cdots\!89}a^{11}+\frac{11\!\cdots\!63}{67\!\cdots\!89}a^{9}+\frac{11\!\cdots\!39}{67\!\cdots\!89}a^{7}+\frac{19\!\cdots\!19}{12\!\cdots\!13}a^{5}+\frac{61\!\cdots\!60}{12\!\cdots\!13}a^{3}+\frac{96\!\cdots\!03}{11\!\cdots\!83}a$, $\frac{21\!\cdots\!33}{20\!\cdots\!86}a^{15}+\frac{25\!\cdots\!65}{19\!\cdots\!81}a^{14}+\frac{74\!\cdots\!14}{19\!\cdots\!81}a^{13}-\frac{44\!\cdots\!28}{19\!\cdots\!81}a^{12}+\frac{17\!\cdots\!21}{39\!\cdots\!62}a^{11}+\frac{20\!\cdots\!13}{36\!\cdots\!77}a^{10}+\frac{18\!\cdots\!73}{13\!\cdots\!78}a^{9}+\frac{22\!\cdots\!08}{12\!\cdots\!13}a^{8}+\frac{80\!\cdots\!33}{39\!\cdots\!62}a^{7}+\frac{18\!\cdots\!41}{73\!\cdots\!54}a^{6}+\frac{29\!\cdots\!24}{36\!\cdots\!77}a^{5}+\frac{32\!\cdots\!77}{36\!\cdots\!77}a^{4}+\frac{29\!\cdots\!83}{36\!\cdots\!77}a^{3}+\frac{41\!\cdots\!15}{73\!\cdots\!54}a^{2}+\frac{79\!\cdots\!76}{33\!\cdots\!07}a+\frac{23\!\cdots\!97}{30\!\cdots\!37}$, $\frac{21\!\cdots\!33}{20\!\cdots\!86}a^{15}-\frac{25\!\cdots\!65}{19\!\cdots\!81}a^{14}+\frac{74\!\cdots\!14}{19\!\cdots\!81}a^{13}+\frac{44\!\cdots\!28}{19\!\cdots\!81}a^{12}+\frac{17\!\cdots\!21}{39\!\cdots\!62}a^{11}-\frac{20\!\cdots\!13}{36\!\cdots\!77}a^{10}+\frac{18\!\cdots\!73}{13\!\cdots\!78}a^{9}-\frac{22\!\cdots\!08}{12\!\cdots\!13}a^{8}+\frac{80\!\cdots\!33}{39\!\cdots\!62}a^{7}-\frac{18\!\cdots\!41}{73\!\cdots\!54}a^{6}+\frac{29\!\cdots\!24}{36\!\cdots\!77}a^{5}-\frac{32\!\cdots\!77}{36\!\cdots\!77}a^{4}+\frac{29\!\cdots\!83}{36\!\cdots\!77}a^{3}-\frac{41\!\cdots\!15}{73\!\cdots\!54}a^{2}+\frac{79\!\cdots\!76}{33\!\cdots\!07}a-\frac{23\!\cdots\!97}{30\!\cdots\!37}$, $\frac{23\!\cdots\!19}{71\!\cdots\!34}a^{15}+\frac{56\!\cdots\!91}{19\!\cdots\!81}a^{14}-\frac{19\!\cdots\!17}{13\!\cdots\!78}a^{13}-\frac{46\!\cdots\!75}{19\!\cdots\!81}a^{12}+\frac{94\!\cdots\!01}{67\!\cdots\!89}a^{11}+\frac{48\!\cdots\!41}{39\!\cdots\!62}a^{10}+\frac{58\!\cdots\!43}{13\!\cdots\!78}a^{9}+\frac{25\!\cdots\!94}{67\!\cdots\!89}a^{8}+\frac{82\!\cdots\!01}{13\!\cdots\!78}a^{7}+\frac{38\!\cdots\!71}{73\!\cdots\!54}a^{6}+\frac{52\!\cdots\!23}{25\!\cdots\!26}a^{5}+\frac{11\!\cdots\!37}{73\!\cdots\!54}a^{4}+\frac{26\!\cdots\!87}{25\!\cdots\!26}a^{3}+\frac{14\!\cdots\!69}{73\!\cdots\!54}a^{2}+\frac{47\!\cdots\!85}{11\!\cdots\!83}a+\frac{27\!\cdots\!72}{30\!\cdots\!37}$, $\frac{12\!\cdots\!08}{10\!\cdots\!93}a^{15}-\frac{18\!\cdots\!52}{19\!\cdots\!81}a^{13}+\frac{10\!\cdots\!56}{19\!\cdots\!81}a^{11}+\frac{10\!\cdots\!87}{67\!\cdots\!89}a^{9}+\frac{42\!\cdots\!94}{19\!\cdots\!81}a^{7}+\frac{26\!\cdots\!34}{36\!\cdots\!77}a^{5}+\frac{18\!\cdots\!20}{36\!\cdots\!77}a^{3}+\frac{30\!\cdots\!16}{33\!\cdots\!07}a$ 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:  \( 22284020753.2 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 22284020753.2 \cdot 30}{2\cdot\sqrt{32858296023668216937367870328998288009}}\cr\approx \mathstrut & 0.141644997547 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 424*x^12 + 130963*x^10 + 1892100*x^8 + 7036545*x^6 + 5578674*x^4 + 1806187*x^2 + 339889)
 
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 + 424*x^12 + 130963*x^10 + 1892100*x^8 + 7036545*x^6 + 5578674*x^4 + 1806187*x^2 + 339889, 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 + 424*x^12 + 130963*x^10 + 1892100*x^8 + 7036545*x^6 + 5578674*x^4 + 1806187*x^2 + 339889);
 
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 + 424*x^12 + 130963*x^10 + 1892100*x^8 + 7036545*x^6 + 5578674*x^4 + 1806187*x^2 + 339889);
 
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

$\OD_{16}:C_2$ (as 16T36):

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 $\OD_{16}:C_2$
Character table for $\OD_{16}:C_2$

Intermediate fields

\(\Q(\sqrt{53}) \), \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{-2491}) \), 4.4.328869293.1, 4.0.148877.1, \(\Q(\sqrt{-47}, \sqrt{53})\), 8.4.5732215629550951997.1 x2, 8.0.2594936907899933.1 x2, 8.0.108155011878319849.2

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: 8.4.5732215629550951997.1, 8.0.2594936907899933.1
Degree 16 siblings: 16.0.14874737901162615182149330162516201.1, 16.4.32858296023668216937367870328998288009.1
Minimal sibling: 8.0.2594936907899933.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ R R ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(47\) Copy content Toggle raw display 47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.4.2.1$x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(53\) Copy content Toggle raw display 53.8.7.2$x^{8} + 53$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.2$x^{8} + 53$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$