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Properties

Label 16.0.214...329.1
Degree $16$
Signature $[0, 8]$
Discriminant $2.146\times 10^{35}$
Root discriminant \(161.52\)
Ramified primes $37,47$
Class number $250$ (GRH)
Class group [5, 5, 10] (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 - 74*x^14 + 777*x^12 + 74037*x^10 + 1059384*x^8 + 9339318*x^6 + 55276113*x^4 + 60835622*x^2 + 6901129)
 
gp: K = bnfinit(y^16 - 74*y^14 + 777*y^12 + 74037*y^10 + 1059384*y^8 + 9339318*y^6 + 55276113*y^4 + 60835622*y^2 + 6901129, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 74*x^14 + 777*x^12 + 74037*x^10 + 1059384*x^8 + 9339318*x^6 + 55276113*x^4 + 60835622*x^2 + 6901129);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 74*x^14 + 777*x^12 + 74037*x^10 + 1059384*x^8 + 9339318*x^6 + 55276113*x^4 + 60835622*x^2 + 6901129)
 

\( x^{16} - 74 x^{14} + 777 x^{12} + 74037 x^{10} + 1059384 x^{8} + 9339318 x^{6} + 55276113 x^{4} + \cdots + 6901129 \) 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:   \(214588774932298492495221177464492329\) \(\medspace = 37^{14}\cdot 47^{8}\) 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:  \(161.52\)
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:  $37^{7/8}47^{1/2}\approx 161.51967702721538$
Ramified primes:   \(37\), \(47\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{74}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{74}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{74}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{5254}a^{11}+\frac{6}{2627}a^{9}-\frac{3}{142}a^{7}-\frac{28}{71}a^{5}+\frac{20}{71}a^{3}-\frac{1}{2}a^{2}+\frac{67}{142}a-\frac{1}{2}$, $\frac{1}{15762}a^{12}+\frac{2}{2627}a^{10}+\frac{31}{15762}a^{8}+\frac{43}{213}a^{6}+\frac{91}{213}a^{4}-\frac{1}{2}a^{3}-\frac{25}{142}a^{2}-\frac{1}{2}a-\frac{1}{3}$, $\frac{1}{15762}a^{13}+\frac{50}{7881}a^{9}-\frac{91}{426}a^{7}-\frac{1}{2}a^{6}+\frac{1}{213}a^{5}-\frac{1}{2}a^{4}+\frac{14}{71}a^{3}-\frac{1}{2}a^{2}+\frac{119}{426}a$, $\frac{1}{15\!\cdots\!02}a^{14}+\frac{22\!\cdots\!54}{78\!\cdots\!01}a^{12}+\frac{32\!\cdots\!77}{15\!\cdots\!02}a^{10}+\frac{30\!\cdots\!47}{15\!\cdots\!02}a^{8}+\frac{16\!\cdots\!65}{14\!\cdots\!82}a^{6}+\frac{20\!\cdots\!41}{42\!\cdots\!46}a^{4}-\frac{1}{2}a^{3}+\frac{25\!\cdots\!43}{42\!\cdots\!46}a^{2}-\frac{40\!\cdots\!13}{83\!\cdots\!06}$, $\frac{1}{57\!\cdots\!74}a^{15}-\frac{68\!\cdots\!15}{15\!\cdots\!02}a^{13}-\frac{65\!\cdots\!26}{78\!\cdots\!01}a^{11}-\frac{13\!\cdots\!08}{78\!\cdots\!01}a^{9}+\frac{83\!\cdots\!44}{26\!\cdots\!67}a^{7}+\frac{11\!\cdots\!25}{42\!\cdots\!46}a^{5}+\frac{87\!\cdots\!83}{21\!\cdots\!73}a^{3}+\frac{35\!\cdots\!80}{29\!\cdots\!63}a-\frac{1}{2}$ 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

$C_{5}\times C_{5}\times C_{10}$, which has order $250$ (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:   $\frac{264353602}{31\!\cdots\!17}a^{14}-\frac{33271877520}{31\!\cdots\!17}a^{12}+\frac{37741463564}{85574250142741}a^{10}-\frac{172774440480}{85574250142741}a^{8}-\frac{10874312645574}{85574250142741}a^{6}-\frac{87612414494696}{85574250142741}a^{4}-\frac{10\!\cdots\!24}{85574250142741}a^{2}-\frac{27962916921}{16975649701}$, $\frac{44\!\cdots\!95}{96\!\cdots\!79}a^{15}-\frac{24\!\cdots\!90}{78\!\cdots\!01}a^{13}+\frac{35\!\cdots\!71}{26\!\cdots\!67}a^{11}+\frac{30\!\cdots\!95}{78\!\cdots\!01}a^{9}+\frac{46\!\cdots\!51}{78\!\cdots\!01}a^{7}+\frac{10\!\cdots\!17}{21\!\cdots\!73}a^{5}+\frac{22\!\cdots\!15}{70\!\cdots\!91}a^{3}+\frac{12\!\cdots\!98}{29\!\cdots\!63}a$, $\frac{240039222459911}{28\!\cdots\!37}a^{15}-\frac{517676554101845}{78\!\cdots\!01}a^{13}+\frac{75\!\cdots\!50}{78\!\cdots\!01}a^{11}+\frac{47\!\cdots\!13}{78\!\cdots\!01}a^{9}+\frac{11\!\cdots\!06}{26\!\cdots\!67}a^{7}+\frac{21\!\cdots\!58}{21\!\cdots\!73}a^{5}+\frac{35\!\cdots\!76}{21\!\cdots\!73}a^{3}+\frac{33\!\cdots\!25}{29\!\cdots\!63}a$, $\frac{833767346430224}{26\!\cdots\!67}a^{14}-\frac{21\!\cdots\!44}{78\!\cdots\!01}a^{12}+\frac{15\!\cdots\!30}{26\!\cdots\!67}a^{10}+\frac{12\!\cdots\!52}{78\!\cdots\!01}a^{8}+\frac{27\!\cdots\!14}{21\!\cdots\!73}a^{6}+\frac{26\!\cdots\!69}{21\!\cdots\!73}a^{4}+\frac{10\!\cdots\!77}{70\!\cdots\!91}a^{2}+\frac{67\!\cdots\!96}{41\!\cdots\!53}$, $\frac{19237095754299}{52\!\cdots\!34}a^{15}+\frac{49238056240129}{26\!\cdots\!67}a^{14}-\frac{12\!\cdots\!39}{52\!\cdots\!34}a^{13}-\frac{37\!\cdots\!15}{26\!\cdots\!67}a^{12}+\frac{374684179461578}{26\!\cdots\!67}a^{11}+\frac{42\!\cdots\!33}{26\!\cdots\!67}a^{10}+\frac{17\!\cdots\!85}{52\!\cdots\!34}a^{9}+\frac{20\!\cdots\!21}{14\!\cdots\!82}a^{8}+\frac{81\!\cdots\!71}{14\!\cdots\!82}a^{7}+\frac{10\!\cdots\!02}{70\!\cdots\!91}a^{6}+\frac{18\!\cdots\!24}{70\!\cdots\!91}a^{5}+\frac{20\!\cdots\!44}{70\!\cdots\!91}a^{4}-\frac{14\!\cdots\!69}{14\!\cdots\!82}a^{3}+\frac{95\!\cdots\!41}{14\!\cdots\!82}a^{2}-\frac{75\!\cdots\!49}{19\!\cdots\!42}a+\frac{906161739151035}{13\!\cdots\!51}$, $\frac{44\!\cdots\!78}{28\!\cdots\!37}a^{15}+\frac{757325845509983}{26\!\cdots\!67}a^{14}-\frac{45\!\cdots\!42}{26\!\cdots\!67}a^{13}-\frac{55\!\cdots\!91}{52\!\cdots\!34}a^{12}+\frac{50\!\cdots\!12}{78\!\cdots\!01}a^{11}-\frac{41\!\cdots\!03}{52\!\cdots\!34}a^{10}+\frac{23\!\cdots\!15}{15\!\cdots\!02}a^{9}+\frac{33\!\cdots\!06}{70\!\cdots\!91}a^{8}-\frac{16\!\cdots\!26}{78\!\cdots\!01}a^{7}+\frac{11\!\cdots\!73}{14\!\cdots\!82}a^{6}-\frac{88\!\cdots\!79}{70\!\cdots\!91}a^{5}+\frac{21\!\cdots\!05}{14\!\cdots\!82}a^{4}-\frac{28\!\cdots\!63}{21\!\cdots\!73}a^{3}+\frac{56\!\cdots\!28}{70\!\cdots\!91}a^{2}-\frac{15\!\cdots\!79}{99\!\cdots\!21}a+\frac{11\!\cdots\!36}{13\!\cdots\!51}$, $\frac{44\!\cdots\!78}{28\!\cdots\!37}a^{15}-\frac{757325845509983}{26\!\cdots\!67}a^{14}-\frac{45\!\cdots\!42}{26\!\cdots\!67}a^{13}+\frac{55\!\cdots\!91}{52\!\cdots\!34}a^{12}+\frac{50\!\cdots\!12}{78\!\cdots\!01}a^{11}+\frac{41\!\cdots\!03}{52\!\cdots\!34}a^{10}+\frac{23\!\cdots\!15}{15\!\cdots\!02}a^{9}-\frac{33\!\cdots\!06}{70\!\cdots\!91}a^{8}-\frac{16\!\cdots\!26}{78\!\cdots\!01}a^{7}-\frac{11\!\cdots\!73}{14\!\cdots\!82}a^{6}-\frac{88\!\cdots\!79}{70\!\cdots\!91}a^{5}-\frac{21\!\cdots\!05}{14\!\cdots\!82}a^{4}-\frac{28\!\cdots\!63}{21\!\cdots\!73}a^{3}-\frac{56\!\cdots\!28}{70\!\cdots\!91}a^{2}-\frac{15\!\cdots\!79}{99\!\cdots\!21}a-\frac{11\!\cdots\!36}{13\!\cdots\!51}$ 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:  \( 339086333.465 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 339086333.465 \cdot 250}{2\cdot\sqrt{214588774932298492495221177464492329}}\cr\approx \mathstrut & 0.222257152504 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 74*x^14 + 777*x^12 + 74037*x^10 + 1059384*x^8 + 9339318*x^6 + 55276113*x^4 + 60835622*x^2 + 6901129)
 
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 - 74*x^14 + 777*x^12 + 74037*x^10 + 1059384*x^8 + 9339318*x^6 + 55276113*x^4 + 60835622*x^2 + 6901129, 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 - 74*x^14 + 777*x^12 + 74037*x^10 + 1059384*x^8 + 9339318*x^6 + 55276113*x^4 + 60835622*x^2 + 6901129);
 
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 - 74*x^14 + 777*x^12 + 74037*x^10 + 1059384*x^8 + 9339318*x^6 + 55276113*x^4 + 60835622*x^2 + 6901129);
 
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

\(\Q(\sqrt{37}) \), \(\Q(\sqrt{-1739}) \), \(\Q(\sqrt{-47}) \), 4.4.111892477.1, 4.0.50653.1, \(\Q(\sqrt{37}, \sqrt{-47})\), 8.4.463237277140234573.1 x2, 8.0.209704516586797.1 x2, 8.0.12519926409195529.1

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: 8.4.463237277140234573.1, 8.0.209704516586797.1
Degree 16 siblings: 16.4.214588774932298492495221177464492329.1, 16.0.97142949267676999771489894732681.1
Minimal sibling: 8.0.209704516586797.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ R ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ R ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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
\(37\) Copy content Toggle raw display 37.8.7.2$x^{8} + 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.2$x^{8} + 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
\(47\) 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}$