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)
\( x^{16} - 3 x^{15} + 6 x^{14} - 4 x^{13} - x^{12} + 15 x^{11} - 27 x^{10} + 32 x^{9} - 12 x^{8} + \cdots + 1 \)
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: |
\(731086699811838561\)
\(\medspace = 3^{16}\cdot 19^{8}\)
| 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))
| |
| Galois root discriminant: | $3^{4/3}19^{1/2}\approx 18.859860385004858$ | ||
| Ramified primes: |
\(3\), \(19\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{56195686}a^{15}-\frac{5048488}{28097843}a^{14}+\frac{1111075}{56195686}a^{13}+\frac{19107259}{56195686}a^{12}+\frac{15673121}{56195686}a^{11}-\frac{9795169}{56195686}a^{10}-\frac{12554447}{56195686}a^{9}-\frac{8559929}{28097843}a^{8}-\frac{983625}{56195686}a^{7}+\frac{1446632}{28097843}a^{6}-\frac{10805966}{28097843}a^{5}-\frac{4099715}{56195686}a^{4}+\frac{7014238}{28097843}a^{3}-\frac{5619997}{28097843}a^{2}-\frac{25527443}{56195686}a+\frac{7866912}{28097843}$
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 $2$)
| 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: |
$\frac{7934539}{28097843}a^{15}-\frac{36032205}{56195686}a^{14}+\frac{29188960}{28097843}a^{13}-\frac{1849667}{56195686}a^{12}-\frac{52396327}{56195686}a^{11}+\frac{197843763}{56195686}a^{10}-\frac{276757383}{56195686}a^{9}+\frac{192830397}{56195686}a^{8}+\frac{42632706}{28097843}a^{7}-\frac{438318261}{56195686}a^{6}+\frac{278020437}{28097843}a^{5}-\frac{166639012}{28097843}a^{4}+\frac{191243773}{56195686}a^{3}+\frac{12604128}{28097843}a^{2}-\frac{48883008}{28097843}a+\frac{54187467}{56195686}$, $\frac{1439675}{28097843}a^{15}-\frac{1042436}{28097843}a^{14}+\frac{4796478}{28097843}a^{13}+\frac{5238337}{28097843}a^{12}+\frac{2871781}{28097843}a^{11}+\frac{26003980}{28097843}a^{10}+\frac{7394827}{28097843}a^{9}+\frac{31041491}{28097843}a^{8}+\frac{30965325}{28097843}a^{7}-\frac{4886335}{28097843}a^{6}+\frac{44439636}{28097843}a^{5}+\frac{3805798}{28097843}a^{4}+\frac{25713173}{28097843}a^{3}+\frac{2791552}{28097843}a^{2}+\frac{26599057}{28097843}a+\frac{11171576}{28097843}$, $\frac{4121951}{28097843}a^{15}-\frac{12822501}{28097843}a^{14}+\frac{16885383}{28097843}a^{13}+\frac{4184490}{28097843}a^{12}-\frac{44445150}{28097843}a^{11}+\frac{73137660}{28097843}a^{10}-\frac{90684178}{28097843}a^{9}+\frac{11130368}{28097843}a^{8}+\frac{120888211}{28097843}a^{7}-\frac{265121129}{28097843}a^{6}+\frac{212163936}{28097843}a^{5}+\frac{5175839}{28097843}a^{4}-\frac{192553993}{28097843}a^{3}+\frac{165988522}{28097843}a^{2}-\frac{99885883}{28097843}a+\frac{43348017}{28097843}$, $\frac{8124973}{28097843}a^{15}-\frac{19902589}{28097843}a^{14}+\frac{38887720}{28097843}a^{13}-\frac{13881849}{28097843}a^{12}-\frac{10397403}{28097843}a^{11}+\frac{111260698}{28097843}a^{10}-\frac{156722427}{28097843}a^{9}+\frac{184442567}{28097843}a^{8}-\frac{28984792}{28097843}a^{7}-\frac{171779177}{28097843}a^{6}+\frac{319111060}{28097843}a^{5}-\frac{305588939}{28097843}a^{4}+\frac{234144010}{28097843}a^{3}-\frac{139320586}{28097843}a^{2}+\frac{62539061}{28097843}a-\frac{1599563}{28097843}$, $\frac{10661733}{56195686}a^{15}-\frac{26264597}{28097843}a^{14}+\frac{102971233}{56195686}a^{13}-\frac{117173647}{56195686}a^{12}-\frac{17575617}{56195686}a^{11}+\frac{192924721}{56195686}a^{10}-\frac{556524227}{56195686}a^{9}+\frac{308198797}{28097843}a^{8}-\frac{413961979}{56195686}a^{7}-\frac{172922577}{28097843}a^{6}+\frac{525756757}{28097843}a^{5}-\frac{1230722353}{56195686}a^{4}+\frac{455891920}{28097843}a^{3}-\frac{257277301}{28097843}a^{2}+\frac{164996481}{56195686}a-\frac{12056490}{28097843}$, $\frac{7970339}{56195686}a^{15}+\frac{1521107}{28097843}a^{14}-\frac{8969571}{56195686}a^{13}+\frac{67204139}{56195686}a^{12}-\frac{244309}{56195686}a^{11}+\frac{50805557}{56195686}a^{10}+\frac{132444845}{56195686}a^{9}-\frac{67541754}{28097843}a^{8}+\frac{260237729}{56195686}a^{7}+\frac{888297}{28097843}a^{6}-\frac{102301823}{28097843}a^{5}+\frac{380554837}{56195686}a^{4}-\frac{86356302}{28097843}a^{3}+\frac{47161030}{28097843}a^{2}-\frac{48998833}{56195686}a-\frac{10806226}{28097843}$, $\frac{7934539}{28097843}a^{15}-\frac{36032205}{56195686}a^{14}+\frac{29188960}{28097843}a^{13}-\frac{1849667}{56195686}a^{12}-\frac{52396327}{56195686}a^{11}+\frac{197843763}{56195686}a^{10}-\frac{276757383}{56195686}a^{9}+\frac{192830397}{56195686}a^{8}+\frac{42632706}{28097843}a^{7}-\frac{438318261}{56195686}a^{6}+\frac{278020437}{28097843}a^{5}-\frac{166639012}{28097843}a^{4}+\frac{191243773}{56195686}a^{3}+\frac{12604128}{28097843}a^{2}-\frac{48883008}{28097843}a-\frac{2008219}{56195686}$
| 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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 3.12.16.14 | $x^{12} + 24 x^{11} + 216 x^{10} + 768 x^{9} - 432 x^{8} - 10368 x^{7} - 18414 x^{6} + 27864 x^{5} + 83592 x^{4} + 10800 x^{3} + 64800 x^{2} + 901125$ | $3$ | $4$ | $16$ | $C_{12}$ | $[2]^{4}$ | |
|
\(19\)
| 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 *.