Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 1)
gp: K = bnfinit(y^24 - 3*y^16 - 8*y^12 + 18*y^8 + 8*y^4 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 1)
\( x^{24} - 3x^{16} - 8x^{12} + 18x^{8} + 8x^{4} + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(3852179415897489839182437154816\)
\(\medspace = 2^{72}\cdot 13^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.81\) | 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: | $2^{3}13^{2/3}\approx 44.23019850943098$ | ||
| Ramified primes: |
\(2\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $12$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{899}a^{20}+\frac{287}{899}a^{16}-\frac{342}{899}a^{12}-\frac{171}{899}a^{8}+\frac{386}{899}a^{4}+\frac{213}{899}$, $\frac{1}{899}a^{21}+\frac{287}{899}a^{17}-\frac{342}{899}a^{13}-\frac{171}{899}a^{9}+\frac{386}{899}a^{5}+\frac{213}{899}a$, $\frac{1}{899}a^{22}+\frac{287}{899}a^{18}-\frac{342}{899}a^{14}-\frac{171}{899}a^{10}+\frac{386}{899}a^{6}+\frac{213}{899}a^{2}$, $\frac{1}{899}a^{23}+\frac{287}{899}a^{19}-\frac{342}{899}a^{15}-\frac{171}{899}a^{11}+\frac{386}{899}a^{7}+\frac{213}{899}a^{3}$
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 $1$ (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: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{952}{899} a^{23} - \frac{72}{899} a^{19} - \frac{2843}{899} a^{15} - \frac{7265}{899} a^{11} + \frac{17761}{899} a^{7} + \frac{5895}{899} a^{3} \)
(order $16$)
| 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: |
$\frac{313}{899}a^{20}-\frac{69}{899}a^{16}-\frac{964}{899}a^{12}-\frac{2280}{899}a^{8}+\frac{5746}{899}a^{4}+\frac{1042}{899}$, $\frac{313}{899}a^{20}-\frac{69}{899}a^{16}-\frac{964}{899}a^{12}-\frac{2280}{899}a^{8}+\frac{6645}{899}a^{4}+\frac{1042}{899}$, $\frac{989}{899}a^{23}+\frac{573}{899}a^{21}-\frac{241}{899}a^{19}-\frac{66}{899}a^{17}-\frac{2911}{899}a^{15}-\frac{1782}{899}a^{13}-\frac{7299}{899}a^{11}-\frac{4487}{899}a^{9}+\frac{19457}{899}a^{7}+\frac{10812}{899}a^{5}+\frac{2988}{899}a^{3}+\frac{3381}{899}a$, $\frac{1042}{899}a^{22}+\frac{138}{899}a^{20}-\frac{313}{899}a^{18}+\frac{50}{899}a^{16}-\frac{3057}{899}a^{14}-\frac{448}{899}a^{12}-\frac{7372}{899}a^{10}-\frac{1123}{899}a^{8}+\frac{21036}{899}a^{6}+\frac{2025}{899}a^{4}+\frac{2590}{899}a^{2}+\frac{1525}{899}$, $\frac{53}{899}a^{22}-\frac{313}{899}a^{20}-\frac{72}{899}a^{18}+\frac{69}{899}a^{16}-\frac{146}{899}a^{14}+\frac{964}{899}a^{12}-\frac{73}{899}a^{10}+\frac{2280}{899}a^{8}+\frac{1579}{899}a^{6}-\frac{5746}{899}a^{4}-\frac{1297}{899}a^{2}-\frac{1042}{899}$, $\frac{814}{899}a^{22}+\frac{626}{899}a^{21}+\frac{15}{31}a^{20}-\frac{122}{899}a^{18}-\frac{138}{899}a^{17}-\frac{4}{31}a^{16}-\frac{2395}{899}a^{14}-\frac{1928}{899}a^{13}-\frac{46}{31}a^{12}-\frac{6142}{899}a^{10}-\frac{4560}{899}a^{9}-\frac{116}{31}a^{8}+\frac{15736}{899}a^{6}+\frac{12391}{899}a^{5}+\frac{303}{31}a^{4}+\frac{4370}{899}a^{2}+\frac{2983}{899}a+\frac{64}{31}$, $\frac{952}{899}a^{23}-\frac{488}{899}a^{22}-\frac{15}{31}a^{20}-\frac{72}{899}a^{19}+\frac{188}{899}a^{18}+\frac{4}{31}a^{16}-\frac{2843}{899}a^{15}+\frac{1480}{899}a^{14}+\frac{46}{31}a^{12}-\frac{7265}{899}a^{11}+\frac{3437}{899}a^{10}+\frac{116}{31}a^{8}+\frac{17761}{899}a^{7}-\frac{10366}{899}a^{6}-\frac{303}{31}a^{4}+\frac{5895}{899}a^{3}-\frac{559}{899}a^{2}-\frac{64}{31}$, $\frac{1127}{899}a^{23}-\frac{814}{899}a^{22}+\frac{501}{899}a^{21}+\frac{363}{899}a^{20}-\frac{191}{899}a^{19}+\frac{122}{899}a^{18}-\frac{53}{899}a^{17}-\frac{103}{899}a^{16}-\frac{3359}{899}a^{15}+\frac{2395}{899}a^{14}-\frac{1431}{899}a^{13}-\frac{983}{899}a^{12}-\frac{8422}{899}a^{11}+\frac{6142}{899}a^{10}-\frac{3862}{899}a^{9}-\frac{2739}{899}a^{8}+\frac{21482}{899}a^{7}-\frac{15736}{899}a^{6}+\frac{9091}{899}a^{5}+\frac{7066}{899}a^{4}+\frac{5412}{899}a^{3}-\frac{3471}{899}a^{2}+\frac{3328}{899}a+\frac{904}{899}$, $\frac{814}{899}a^{23}+\frac{53}{899}a^{21}+\frac{122}{899}a^{20}-\frac{122}{899}a^{19}-\frac{72}{899}a^{17}-\frac{47}{899}a^{16}-\frac{2395}{899}a^{15}-\frac{146}{899}a^{13}-\frac{370}{899}a^{12}-\frac{6142}{899}a^{11}-\frac{73}{899}a^{9}-\frac{1084}{899}a^{8}+\frac{15736}{899}a^{7}+\frac{1579}{899}a^{5}+\frac{2142}{899}a^{4}+\frac{4370}{899}a^{3}-a^{2}-\frac{398}{899}a-\frac{85}{899}$, $\frac{90}{899}a^{23}+\frac{1042}{899}a^{22}+\frac{814}{899}a^{21}+\frac{326}{899}a^{20}-\frac{241}{899}a^{19}-\frac{313}{899}a^{18}-\frac{122}{899}a^{17}+\frac{66}{899}a^{16}-\frac{214}{899}a^{15}-\frac{3057}{899}a^{14}-\frac{2395}{899}a^{13}-\frac{915}{899}a^{12}-\frac{107}{899}a^{11}-\frac{7372}{899}a^{10}-\frac{6142}{899}a^{9}-\frac{2705}{899}a^{8}+\frac{3275}{899}a^{7}+\frac{21036}{899}a^{6}+\frac{15736}{899}a^{5}+\frac{5370}{899}a^{4}-\frac{4204}{899}a^{3}+\frac{2590}{899}a^{2}+\frac{4370}{899}a+\frac{2912}{899}$, $\frac{573}{899}a^{23}+\frac{379}{899}a^{22}-\frac{379}{899}a^{21}+\frac{379}{899}a^{20}-\frac{66}{899}a^{19}-\frac{6}{899}a^{18}+\frac{6}{899}a^{17}-\frac{6}{899}a^{16}-\frac{1782}{899}a^{15}-\frac{1061}{899}a^{14}+\frac{1061}{899}a^{13}-\frac{1061}{899}a^{12}-\frac{4487}{899}a^{11}-\frac{2778}{899}a^{10}+\frac{2778}{899}a^{9}-\frac{2778}{899}a^{8}+\frac{10812}{899}a^{7}+\frac{6949}{899}a^{6}-\frac{6949}{899}a^{5}+\frac{6949}{899}a^{4}+\frac{4280}{899}a^{3}+\frac{1615}{899}a^{2}-\frac{1615}{899}a+\frac{1615}{899}$
| 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: | \( 2446825.407448486 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 2446825.407448486 \cdot 1}{16\cdot\sqrt{3852179415897489839182437154816}}\cr\approx \mathstrut & 0.294977007875353 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 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^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 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^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 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^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 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
$S_3\times C_{12}$ (as 24T65):
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 72 |
| The 36 conjugacy class representatives for $S_3\times C_{12}$ |
| Character table for $S_3\times C_{12}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), 4.0.2048.2, \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{8})\), 6.0.10816.1, \(\Q(\zeta_{16})\), 12.0.479174066176.4 |
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 36 siblings: | data not computed |
| 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 | R | ${\href{/padicField/3.12.0.1}{12} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{4}$ | ${\href{/padicField/11.12.0.1}{12} }^{2}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{12}$ | ${\href{/padicField/19.12.0.1}{12} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{12}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.12.0.1}{12} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{12}$ | ${\href{/padicField/53.12.0.1}{12} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $24$ | $8$ | $3$ | $72$ | |||
|
\(13\)
| 13.12.8.2 | $x^{12} + 507 x^{6} - 26364 x^{3} + 57122$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
| 13.12.0.1 | $x^{12} + x^{8} + 5 x^{7} + 8 x^{6} + 11 x^{5} + 3 x^{4} + x^{3} + x^{2} + 4 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.52.6t1.b.a | $1$ | $ 2^{2} \cdot 13 $ | 6.0.1827904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.52.6t1.b.b | $1$ | $ 2^{2} \cdot 13 $ | 6.0.1827904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.104.6t1.d.a | $1$ | $ 2^{3} \cdot 13 $ | 6.0.14623232.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.13.3t1.a.a | $1$ | $ 13 $ | 3.3.169.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.13.3t1.a.b | $1$ | $ 13 $ | 3.3.169.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.104.6t1.a.a | $1$ | $ 2^{3} \cdot 13 $ | 6.6.14623232.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.104.6t1.a.b | $1$ | $ 2^{3} \cdot 13 $ | 6.6.14623232.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.104.6t1.d.b | $1$ | $ 2^{3} \cdot 13 $ | 6.0.14623232.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.16.4t1.a.a | $1$ | $ 2^{4}$ | \(\Q(\zeta_{16})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.16.4t1.b.a | $1$ | $ 2^{4}$ | 4.0.2048.2 | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.16.4t1.b.b | $1$ | $ 2^{4}$ | 4.0.2048.2 | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.16.4t1.a.b | $1$ | $ 2^{4}$ | \(\Q(\zeta_{16})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| 1.208.12t1.b.a | $1$ | $ 2^{4} \cdot 13 $ | 12.12.7007073538075000832.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.208.12t1.a.a | $1$ | $ 2^{4} \cdot 13 $ | 12.0.7007073538075000832.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.208.12t1.a.b | $1$ | $ 2^{4} \cdot 13 $ | 12.0.7007073538075000832.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.208.12t1.b.b | $1$ | $ 2^{4} \cdot 13 $ | 12.12.7007073538075000832.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.208.12t1.b.c | $1$ | $ 2^{4} \cdot 13 $ | 12.12.7007073538075000832.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.208.12t1.b.d | $1$ | $ 2^{4} \cdot 13 $ | 12.12.7007073538075000832.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.208.12t1.a.c | $1$ | $ 2^{4} \cdot 13 $ | 12.0.7007073538075000832.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.208.12t1.a.d | $1$ | $ 2^{4} \cdot 13 $ | 12.0.7007073538075000832.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 2.676.3t2.b.a | $2$ | $ 2^{2} \cdot 13^{2}$ | 3.1.676.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.10816.6t3.d.a | $2$ | $ 2^{6} \cdot 13^{2}$ | 6.0.58492928.3 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.832.12t18.b.a | $2$ | $ 2^{6} \cdot 13 $ | 12.0.479174066176.4 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.832.12t18.b.b | $2$ | $ 2^{6} \cdot 13 $ | 12.0.479174066176.4 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.52.6t5.b.a | $2$ | $ 2^{2} \cdot 13 $ | 6.0.10816.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.52.6t5.b.b | $2$ | $ 2^{2} \cdot 13 $ | 6.0.10816.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| 2.43264.12t11.b.a | $2$ | $ 2^{8} \cdot 13^{2}$ | 12.4.28028294152300003328.26 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| 2.43264.12t11.b.b | $2$ | $ 2^{8} \cdot 13^{2}$ | 12.4.28028294152300003328.26 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| * | 2.3328.24t65.a.a | $2$ | $ 2^{8} \cdot 13 $ | 24.0.3852179415897489839182437154816.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.3328.24t65.a.b | $2$ | $ 2^{8} \cdot 13 $ | 24.0.3852179415897489839182437154816.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.3328.24t65.a.c | $2$ | $ 2^{8} \cdot 13 $ | 24.0.3852179415897489839182437154816.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.3328.24t65.a.d | $2$ | $ 2^{8} \cdot 13 $ | 24.0.3852179415897489839182437154816.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
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 *.