Properties

Label 16.0.731086699811838561.1
Degree $16$
Signature $[0, 8]$
Discriminant $7.311\times 10^{17}$
Root discriminant \(13.08\)
Ramified primes $3,19$
Class number $1$
Class group trivial
Galois group $\SL(2,3):C_2$ (as 16T60)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 6*x^14 - 4*x^13 - x^12 + 15*x^11 - 27*x^10 + 32*x^9 - 12*x^8 - 23*x^7 + 55*x^6 - 57*x^5 + 43*x^4 - 23*x^3 + 9*x^2 - x + 1)
 
gp: K = bnfinit(y^16 - 3*y^15 + 6*y^14 - 4*y^13 - y^12 + 15*y^11 - 27*y^10 + 32*y^9 - 12*y^8 - 23*y^7 + 55*y^6 - 57*y^5 + 43*y^4 - 23*y^3 + 9*y^2 - y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 6*x^14 - 4*x^13 - x^12 + 15*x^11 - 27*x^10 + 32*x^9 - 12*x^8 - 23*x^7 + 55*x^6 - 57*x^5 + 43*x^4 - 23*x^3 + 9*x^2 - x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 6*x^14 - 4*x^13 - x^12 + 15*x^11 - 27*x^10 + 32*x^9 - 12*x^8 - 23*x^7 + 55*x^6 - 57*x^5 + 43*x^4 - 23*x^3 + 9*x^2 - x + 1)
 

\( x^{16} - 3 x^{15} + 6 x^{14} - 4 x^{13} - x^{12} + 15 x^{11} - 27 x^{10} + 32 x^{9} - 12 x^{8} - 23 x^{7} + 55 x^{6} - 57 x^{5} + 43 x^{4} - 23 x^{3} + 9 x^{2} - x + 1 \) 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:   \(731086699811838561\) \(\medspace = 3^{16}\cdot 19^{8}\) 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:  \(13.08\)
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))
 
Ramified primes:   \(3\), \(19\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{56195686}a^{15}-\frac{5048488}{28097843}a^{14}+\frac{1111075}{56195686}a^{13}+\frac{19107259}{56195686}a^{12}+\frac{15673121}{56195686}a^{11}-\frac{9795169}{56195686}a^{10}-\frac{12554447}{56195686}a^{9}-\frac{8559929}{28097843}a^{8}-\frac{983625}{56195686}a^{7}+\frac{1446632}{28097843}a^{6}-\frac{10805966}{28097843}a^{5}-\frac{4099715}{56195686}a^{4}+\frac{7014238}{28097843}a^{3}-\frac{5619997}{28097843}a^{2}-\frac{25527443}{56195686}a+\frac{7866912}{28097843}$ 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 $1$

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:   $\frac{7934539}{28097843}a^{15}-\frac{36032205}{56195686}a^{14}+\frac{29188960}{28097843}a^{13}-\frac{1849667}{56195686}a^{12}-\frac{52396327}{56195686}a^{11}+\frac{197843763}{56195686}a^{10}-\frac{276757383}{56195686}a^{9}+\frac{192830397}{56195686}a^{8}+\frac{42632706}{28097843}a^{7}-\frac{438318261}{56195686}a^{6}+\frac{278020437}{28097843}a^{5}-\frac{166639012}{28097843}a^{4}+\frac{191243773}{56195686}a^{3}+\frac{12604128}{28097843}a^{2}-\frac{48883008}{28097843}a+\frac{54187467}{56195686}$, $\frac{1439675}{28097843}a^{15}-\frac{1042436}{28097843}a^{14}+\frac{4796478}{28097843}a^{13}+\frac{5238337}{28097843}a^{12}+\frac{2871781}{28097843}a^{11}+\frac{26003980}{28097843}a^{10}+\frac{7394827}{28097843}a^{9}+\frac{31041491}{28097843}a^{8}+\frac{30965325}{28097843}a^{7}-\frac{4886335}{28097843}a^{6}+\frac{44439636}{28097843}a^{5}+\frac{3805798}{28097843}a^{4}+\frac{25713173}{28097843}a^{3}+\frac{2791552}{28097843}a^{2}+\frac{26599057}{28097843}a+\frac{11171576}{28097843}$, $\frac{4121951}{28097843}a^{15}-\frac{12822501}{28097843}a^{14}+\frac{16885383}{28097843}a^{13}+\frac{4184490}{28097843}a^{12}-\frac{44445150}{28097843}a^{11}+\frac{73137660}{28097843}a^{10}-\frac{90684178}{28097843}a^{9}+\frac{11130368}{28097843}a^{8}+\frac{120888211}{28097843}a^{7}-\frac{265121129}{28097843}a^{6}+\frac{212163936}{28097843}a^{5}+\frac{5175839}{28097843}a^{4}-\frac{192553993}{28097843}a^{3}+\frac{165988522}{28097843}a^{2}-\frac{99885883}{28097843}a+\frac{43348017}{28097843}$, $\frac{8124973}{28097843}a^{15}-\frac{19902589}{28097843}a^{14}+\frac{38887720}{28097843}a^{13}-\frac{13881849}{28097843}a^{12}-\frac{10397403}{28097843}a^{11}+\frac{111260698}{28097843}a^{10}-\frac{156722427}{28097843}a^{9}+\frac{184442567}{28097843}a^{8}-\frac{28984792}{28097843}a^{7}-\frac{171779177}{28097843}a^{6}+\frac{319111060}{28097843}a^{5}-\frac{305588939}{28097843}a^{4}+\frac{234144010}{28097843}a^{3}-\frac{139320586}{28097843}a^{2}+\frac{62539061}{28097843}a-\frac{1599563}{28097843}$, $\frac{10661733}{56195686}a^{15}-\frac{26264597}{28097843}a^{14}+\frac{102971233}{56195686}a^{13}-\frac{117173647}{56195686}a^{12}-\frac{17575617}{56195686}a^{11}+\frac{192924721}{56195686}a^{10}-\frac{556524227}{56195686}a^{9}+\frac{308198797}{28097843}a^{8}-\frac{413961979}{56195686}a^{7}-\frac{172922577}{28097843}a^{6}+\frac{525756757}{28097843}a^{5}-\frac{1230722353}{56195686}a^{4}+\frac{455891920}{28097843}a^{3}-\frac{257277301}{28097843}a^{2}+\frac{164996481}{56195686}a-\frac{12056490}{28097843}$, $\frac{7970339}{56195686}a^{15}+\frac{1521107}{28097843}a^{14}-\frac{8969571}{56195686}a^{13}+\frac{67204139}{56195686}a^{12}-\frac{244309}{56195686}a^{11}+\frac{50805557}{56195686}a^{10}+\frac{132444845}{56195686}a^{9}-\frac{67541754}{28097843}a^{8}+\frac{260237729}{56195686}a^{7}+\frac{888297}{28097843}a^{6}-\frac{102301823}{28097843}a^{5}+\frac{380554837}{56195686}a^{4}-\frac{86356302}{28097843}a^{3}+\frac{47161030}{28097843}a^{2}-\frac{48998833}{56195686}a-\frac{10806226}{28097843}$, $\frac{7934539}{28097843}a^{15}-\frac{36032205}{56195686}a^{14}+\frac{29188960}{28097843}a^{13}-\frac{1849667}{56195686}a^{12}-\frac{52396327}{56195686}a^{11}+\frac{197843763}{56195686}a^{10}-\frac{276757383}{56195686}a^{9}+\frac{192830397}{56195686}a^{8}+\frac{42632706}{28097843}a^{7}-\frac{438318261}{56195686}a^{6}+\frac{278020437}{28097843}a^{5}-\frac{166639012}{28097843}a^{4}+\frac{191243773}{56195686}a^{3}+\frac{12604128}{28097843}a^{2}-\frac{48883008}{28097843}a-\frac{2008219}{56195686}$ 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:  \( 188.5109795655095 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 188.5109795655095 \cdot 1}{2\cdot\sqrt{731086699811838561}}\cr\approx \mathstrut & 0.267769532159754 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 6*x^14 - 4*x^13 - x^12 + 15*x^11 - 27*x^10 + 32*x^9 - 12*x^8 - 23*x^7 + 55*x^6 - 57*x^5 + 43*x^4 - 23*x^3 + 9*x^2 - x + 1)
 
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^15 + 6*x^14 - 4*x^13 - x^12 + 15*x^11 - 27*x^10 + 32*x^9 - 12*x^8 - 23*x^7 + 55*x^6 - 57*x^5 + 43*x^4 - 23*x^3 + 9*x^2 - x + 1, 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^15 + 6*x^14 - 4*x^13 - x^12 + 15*x^11 - 27*x^10 + 32*x^9 - 12*x^8 - 23*x^7 + 55*x^6 - 57*x^5 + 43*x^4 - 23*x^3 + 9*x^2 - x + 1);
 
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^15 + 6*x^14 - 4*x^13 - x^12 + 15*x^11 - 27*x^10 + 32*x^9 - 12*x^8 - 23*x^7 + 55*x^6 - 57*x^5 + 43*x^4 - 23*x^3 + 9*x^2 - x + 1);
 
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

$\SL(2,3):C_2$ (as 16T60):

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 48
The 14 conjugacy class representatives for $\SL(2,3):C_2$
Character table for $\SL(2,3):C_2$

Intermediate fields

\(\Q(\sqrt{-19}) \), 4.0.29241.1, 8.0.855036081.1

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

Degree 24 sibling: deg 24
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.4.0.1}{4} }$ R ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ R ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }$

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
\(3\) Copy content Toggle raw display 3.4.0.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.12.16.14$x^{12} + 24 x^{11} + 216 x^{10} + 768 x^{9} - 432 x^{8} - 10368 x^{7} - 18414 x^{6} + 27864 x^{5} + 83592 x^{4} + 10800 x^{3} + 64800 x^{2} + 901125$$3$$4$$16$$C_{12}$$[2]^{4}$
\(19\) Copy content Toggle raw display 19.8.4.1$x^{8} + 80 x^{6} + 22 x^{5} + 2250 x^{4} - 792 x^{3} + 25817 x^{2} - 22946 x + 107924$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
19.8.4.1$x^{8} + 80 x^{6} + 22 x^{5} + 2250 x^{4} - 792 x^{3} + 25817 x^{2} - 22946 x + 107924$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.19.2t1.a.a$1$ $ 19 $ \(\Q(\sqrt{-19}) \) $C_2$ (as 2T1) $1$ $-1$
1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.171.6t1.f.a$1$ $ 3^{2} \cdot 19 $ 6.0.45001899.1 $C_6$ (as 6T1) $0$ $-1$
1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.171.6t1.f.b$1$ $ 3^{2} \cdot 19 $ 6.0.45001899.1 $C_6$ (as 6T1) $0$ $-1$
2.1539.24t21.a.a$2$ $ 3^{4} \cdot 19 $ 16.0.731086699811838561.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
2.1539.24t21.a.b$2$ $ 3^{4} \cdot 19 $ 16.0.731086699811838561.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 2.171.16t60.a.a$2$ $ 3^{2} \cdot 19 $ 16.0.731086699811838561.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 2.171.16t60.a.b$2$ $ 3^{2} \cdot 19 $ 16.0.731086699811838561.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 2.171.16t60.a.c$2$ $ 3^{2} \cdot 19 $ 16.0.731086699811838561.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 2.171.16t60.a.d$2$ $ 3^{2} \cdot 19 $ 16.0.731086699811838561.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 3.1539.6t6.a.a$3$ $ 3^{4} \cdot 19 $ 6.4.124659.1 $A_4\times C_2$ (as 6T6) $1$ $1$
* 3.29241.4t4.b.a$3$ $ 3^{4} \cdot 19^{2}$ 4.0.29241.1 $A_4$ (as 4T4) $1$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.