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Properties

Label 16.0.259...696.11
Degree $16$
Signature $[0, 8]$
Discriminant $2.599\times 10^{21}$
Root discriminant \(21.80\)
Ramified primes $2,3,7$
Class number $4$
Class group [2, 2]
Galois group $C_2^2 \times D_4$ (as 16T25)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 + 74*x^12 - 156*x^10 - 35*x^8 + 480*x^6 - 36*x^4 - 216*x^2 + 324)
 
gp: K = bnfinit(y^16 - 12*y^14 + 74*y^12 - 156*y^10 - 35*y^8 + 480*y^6 - 36*y^4 - 216*y^2 + 324, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 12*x^14 + 74*x^12 - 156*x^10 - 35*x^8 + 480*x^6 - 36*x^4 - 216*x^2 + 324);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 12*x^14 + 74*x^12 - 156*x^10 - 35*x^8 + 480*x^6 - 36*x^4 - 216*x^2 + 324)
 

\( x^{16} - 12x^{14} + 74x^{12} - 156x^{10} - 35x^{8} + 480x^{6} - 36x^{4} - 216x^{2} + 324 \) 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:   \(2599167103947239325696\) \(\medspace = 2^{36}\cdot 3^{8}\cdot 7^{8}\) 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:  \(21.80\)
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:  $2^{9/4}3^{1/2}7^{1/2}\approx 21.798526485920096$
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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{12}a^{10}-\frac{1}{4}a^{8}-\frac{1}{12}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{6}a^{2}-\frac{1}{2}$, $\frac{1}{12}a^{11}-\frac{1}{4}a^{9}-\frac{1}{12}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{6}a^{3}-\frac{1}{2}a$, $\frac{1}{36}a^{12}+\frac{1}{18}a^{8}-\frac{1}{6}a^{6}-\frac{1}{2}a^{5}+\frac{1}{36}a^{4}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}-\frac{1}{2}$, $\frac{1}{36}a^{13}+\frac{1}{18}a^{9}-\frac{1}{6}a^{7}-\frac{1}{2}a^{6}+\frac{1}{36}a^{5}-\frac{1}{2}a^{4}-\frac{1}{3}a^{3}-\frac{1}{2}a$, $\frac{1}{1107827604}a^{14}+\frac{138208}{30772989}a^{12}+\frac{12452545}{553913802}a^{10}-\frac{37906129}{184637934}a^{8}+\frac{439707043}{1107827604}a^{6}-\frac{81008885}{184637934}a^{4}+\frac{3233081}{61545978}a^{2}-\frac{2701399}{10257663}$, $\frac{1}{1107827604}a^{15}+\frac{138208}{30772989}a^{13}+\frac{12452545}{553913802}a^{11}-\frac{37906129}{184637934}a^{9}-\frac{114206759}{1107827604}a^{7}-\frac{1}{2}a^{6}+\frac{5655041}{92318967}a^{5}-\frac{1}{2}a^{4}+\frac{3233081}{61545978}a^{3}-\frac{2701399}{10257663}a$ 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:  $3$

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:   \( \frac{212}{195453} a^{14} - \frac{155}{14478} a^{12} + \frac{18929}{390906} a^{10} + \frac{2840}{65151} a^{8} - \frac{250217}{390906} a^{6} + \frac{107021}{130302} a^{4} + \frac{20348}{21717} a^{2} - \frac{4802}{7239} \)  (order $4$) 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:   $\frac{523651}{1107827604}a^{14}-\frac{367777}{61545978}a^{12}+\frac{11022842}{276956901}a^{10}-\frac{19716331}{184637934}a^{8}+\frac{112541359}{1107827604}a^{6}+\frac{13916210}{92318967}a^{4}+\frac{19851233}{61545978}a^{2}+\frac{2985929}{10257663}$, $\frac{647243}{1107827604}a^{14}-\frac{1416253}{123091956}a^{12}+\frac{107295901}{1107827604}a^{10}-\frac{160651645}{369275868}a^{8}+\frac{229448558}{276956901}a^{6}-\frac{45928664}{92318967}a^{4}-\frac{12747316}{30772989}a^{2}+\frac{8353708}{10257663}$, $\frac{290305}{184637934}a^{15}+\frac{16021}{369275868}a^{14}-\frac{1085393}{61545978}a^{13}-\frac{1975}{123091956}a^{12}+\frac{9966439}{92318967}a^{11}+\frac{467879}{92318967}a^{10}-\frac{2270897}{10257663}a^{9}-\frac{1415839}{30772989}a^{8}+\frac{25177105}{184637934}a^{7}+\frac{116985979}{369275868}a^{6}+\frac{538198}{3419221}a^{5}-\frac{29520893}{123091956}a^{4}+\frac{1137063}{6838442}a^{3}-\frac{17684333}{20515326}a^{2}+\frac{2364509}{3419221}a-\frac{647143}{6838442}$, $\frac{290305}{184637934}a^{15}-\frac{1249679}{1107827604}a^{14}-\frac{1085393}{61545978}a^{13}+\frac{1319785}{123091956}a^{12}+\frac{9966439}{92318967}a^{11}-\frac{29629667}{553913802}a^{10}-\frac{2270897}{10257663}a^{9}+\frac{223237}{92318967}a^{8}+\frac{25177105}{184637934}a^{7}+\frac{358157041}{1107827604}a^{6}+\frac{538198}{3419221}a^{5}-\frac{214734835}{369275868}a^{4}+\frac{1137063}{6838442}a^{3}-\frac{4613233}{61545978}a^{2}+\frac{2364509}{3419221}a-\frac{4965029}{20515326}$, $\frac{610585}{184637934}a^{15}+\frac{932809}{184637934}a^{14}-\frac{5033831}{123091956}a^{13}-\frac{4157617}{61545978}a^{12}+\frac{47998355}{184637934}a^{11}+\frac{85647095}{184637934}a^{10}-\frac{18769310}{30772989}a^{9}-\frac{42866597}{30772989}a^{8}+\frac{13141501}{184637934}a^{7}+\frac{287238937}{184637934}a^{6}+\frac{228913727}{123091956}a^{5}+\frac{20212103}{30772989}a^{4}-\frac{16481965}{10257663}a^{3}-\frac{11863322}{10257663}a^{2}+\frac{4265603}{6838442}a+\frac{2708694}{3419221}$, $\frac{49919}{29153358}a^{15}+\frac{2602}{2540889}a^{14}-\frac{72001}{3239262}a^{13}-\frac{7369}{564642}a^{12}+\frac{8766065}{58306716}a^{11}+\frac{211124}{2540889}a^{10}-\frac{8502347}{19435572}a^{9}-\frac{166196}{846963}a^{8}+\frac{29454589}{58306716}a^{7}-\frac{250513}{5081778}a^{6}+\frac{1632641}{19435572}a^{5}+\frac{796253}{846963}a^{4}-\frac{1099021}{3239262}a^{3}-\frac{248792}{282321}a^{2}+\frac{1140655}{1079754}a-\frac{616}{94107}$, $\frac{757007}{553913802}a^{15}-\frac{212}{195453}a^{14}-\frac{71094}{3419221}a^{13}+\frac{155}{14478}a^{12}+\frac{173006681}{1107827604}a^{11}-\frac{18929}{390906}a^{10}-\frac{207233003}{369275868}a^{9}-\frac{2840}{65151}a^{8}+\frac{854472067}{1107827604}a^{7}+\frac{250217}{390906}a^{6}+\frac{221446445}{369275868}a^{5}-\frac{107021}{130302}a^{4}-\frac{119205433}{61545978}a^{3}-\frac{20348}{21717}a^{2}-\frac{10255463}{20515326}a-\frac{2437}{7239}$ Copy content Toggle raw display
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:  \( 25816.0461632 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 25816.0461632 \cdot 4}{4\cdot\sqrt{2599167103947239325696}}\cr\approx \mathstrut & 1.23001830599 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 + 74*x^12 - 156*x^10 - 35*x^8 + 480*x^6 - 36*x^4 - 216*x^2 + 324)
 
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 - 12*x^14 + 74*x^12 - 156*x^10 - 35*x^8 + 480*x^6 - 36*x^4 - 216*x^2 + 324, 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 - 12*x^14 + 74*x^12 - 156*x^10 - 35*x^8 + 480*x^6 - 36*x^4 - 216*x^2 + 324);
 
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 - 12*x^14 + 74*x^12 - 156*x^10 - 35*x^8 + 480*x^6 - 36*x^4 - 216*x^2 + 324);
 
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

$C_2^2\times D_4$ (as 16T25):

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 20 conjugacy class representatives for $C_2^2 \times D_4$
Character table for $C_2^2 \times D_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-42}) \), 4.2.37632.1, 4.2.9408.2, 4.2.37632.2, 4.2.9408.1, \(\Q(\sqrt{6}, \sqrt{-7})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{7})\), \(\Q(i, \sqrt{7})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(i, \sqrt{42})\), \(\Q(\sqrt{-6}, \sqrt{7})\), 8.0.12745506816.7, 8.4.50982027264.2, 8.4.50982027264.1, 8.0.1416167424.1, 8.0.88510464.1, 8.0.50982027264.1, 8.0.50982027264.2

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 16 siblings: 16.0.1082535236962615296.2, 16.0.2599167103947239325696.9, 16.8.2599167103947239325696.2, 16.0.2599167103947239325696.10, 16.0.32088482764780732416.2, 16.0.162447943996702457856.14, 16.0.162447943996702457856.16
Minimal sibling: 16.0.1082535236962615296.2

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 ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(2\) Copy content Toggle raw display 2.8.18.55$x^{8} + 2 x^{6} + 4 x^{3} + 10$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
2.8.18.55$x^{8} + 2 x^{6} + 4 x^{3} + 10$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
\(3\) Copy content Toggle raw display 3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{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}$