Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 31*x^14 - 93*x^13 + 202*x^12 - 300*x^11 + 277*x^10 + 65*x^9 - 605*x^8 + 1100*x^7 - 946*x^6 + 740*x^5 + 640*x^4 - 1463*x^3 + 715*x^2 - 42*x + 28)
gp: K = bnfinit(y^16 - 7*y^15 + 31*y^14 - 93*y^13 + 202*y^12 - 300*y^11 + 277*y^10 + 65*y^9 - 605*y^8 + 1100*y^7 - 946*y^6 + 740*y^5 + 640*y^4 - 1463*y^3 + 715*y^2 - 42*y + 28, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 7*x^15 + 31*x^14 - 93*x^13 + 202*x^12 - 300*x^11 + 277*x^10 + 65*x^9 - 605*x^8 + 1100*x^7 - 946*x^6 + 740*x^5 + 640*x^4 - 1463*x^3 + 715*x^2 - 42*x + 28);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 7*x^15 + 31*x^14 - 93*x^13 + 202*x^12 - 300*x^11 + 277*x^10 + 65*x^9 - 605*x^8 + 1100*x^7 - 946*x^6 + 740*x^5 + 640*x^4 - 1463*x^3 + 715*x^2 - 42*x + 28)
\( x^{16} - 7 x^{15} + 31 x^{14} - 93 x^{13} + 202 x^{12} - 300 x^{11} + 277 x^{10} + 65 x^{9} - 605 x^{8} + \cdots + 28 \)
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: |
\(202428293557942478212641\)
\(\medspace = 3^{16}\cdot 7^{8}\cdot 13^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(28.62\) | 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: | $3^{4/3}7^{1/2}13^{2/3}\approx 63.2906556206417$ | ||
| Ramified primes: |
\(3\), \(7\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7}a^{9}-\frac{3}{7}a^{8}+\frac{2}{7}a^{7}+\frac{2}{7}a^{6}-\frac{2}{7}a^{4}-\frac{3}{7}a^{3}+\frac{3}{7}a^{2}$, $\frac{1}{7}a^{10}+\frac{1}{7}a^{7}-\frac{1}{7}a^{6}-\frac{2}{7}a^{5}-\frac{2}{7}a^{4}+\frac{1}{7}a^{3}+\frac{2}{7}a^{2}$, $\frac{1}{7}a^{11}+\frac{1}{7}a^{8}-\frac{1}{7}a^{7}-\frac{2}{7}a^{6}-\frac{2}{7}a^{5}+\frac{1}{7}a^{4}+\frac{2}{7}a^{3}$, $\frac{1}{14}a^{12}-\frac{1}{14}a^{11}-\frac{1}{14}a^{9}-\frac{3}{14}a^{8}-\frac{5}{14}a^{7}+\frac{3}{14}a^{6}+\frac{3}{14}a^{5}+\frac{5}{14}a^{4}+\frac{2}{7}a^{3}-\frac{3}{7}a^{2}-\frac{1}{2}a$, $\frac{1}{98}a^{13}+\frac{1}{98}a^{12}-\frac{1}{49}a^{11}+\frac{3}{98}a^{10}-\frac{1}{98}a^{9}+\frac{33}{98}a^{8}-\frac{23}{98}a^{7}-\frac{43}{98}a^{6}+\frac{45}{98}a^{5}+\frac{20}{49}a^{4}-\frac{3}{49}a^{3}-\frac{13}{98}a^{2}-\frac{1}{7}a+\frac{2}{7}$, $\frac{1}{98}a^{14}-\frac{3}{98}a^{12}+\frac{5}{98}a^{11}-\frac{2}{49}a^{10}+\frac{3}{49}a^{9}+\frac{2}{7}a^{8}+\frac{11}{49}a^{7}+\frac{16}{49}a^{6}-\frac{5}{98}a^{5}+\frac{5}{49}a^{4}-\frac{3}{14}a^{3}+\frac{13}{98}a^{2}+\frac{3}{7}a-\frac{2}{7}$, $\frac{1}{939697322296894}a^{15}+\frac{894070721433}{939697322296894}a^{14}-\frac{933508545035}{939697322296894}a^{13}+\frac{7548104894534}{469848661148447}a^{12}+\frac{43462240269295}{939697322296894}a^{11}-\frac{29452833215273}{469848661148447}a^{10}-\frac{21819910919944}{469848661148447}a^{9}+\frac{732603463344}{469848661148447}a^{8}+\frac{191401792491741}{469848661148447}a^{7}+\frac{224057766575803}{939697322296894}a^{6}+\frac{135116488645465}{939697322296894}a^{5}-\frac{23240726137759}{939697322296894}a^{4}+\frac{12522761337026}{469848661148447}a^{3}+\frac{137762641591049}{939697322296894}a^{2}+\frac{10167531569917}{67121237306921}a+\frac{18935503098678}{67121237306921}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $7$ |
Class group and class number
$C_{2}$, which has order $2$
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: |
$\frac{3025161870567}{469848661148447}a^{15}-\frac{16692878293430}{469848661148447}a^{14}+\frac{65673108351888}{469848661148447}a^{13}-\frac{326398724479161}{939697322296894}a^{12}+\frac{567311734071881}{939697322296894}a^{11}-\frac{259968139308856}{469848661148447}a^{10}+\frac{84534987519485}{939697322296894}a^{9}+\frac{12\!\cdots\!81}{939697322296894}a^{8}-\frac{13\!\cdots\!67}{939697322296894}a^{7}+\frac{10\!\cdots\!25}{939697322296894}a^{6}+\frac{24\!\cdots\!93}{939697322296894}a^{5}-\frac{195532390256011}{939697322296894}a^{4}+\frac{39\!\cdots\!82}{469848661148447}a^{3}+\frac{545875133779916}{469848661148447}a^{2}-\frac{702341772390263}{134242474613842}a+\frac{253252782875629}{67121237306921}$, $\frac{2546284664435}{134242474613842}a^{15}-\frac{8875227216651}{67121237306921}a^{14}+\frac{78519301310641}{134242474613842}a^{13}-\frac{117425793256290}{67121237306921}a^{12}+\frac{508168516309425}{134242474613842}a^{11}-\frac{374139960816231}{67121237306921}a^{10}+\frac{675202375969959}{134242474613842}a^{9}+\frac{210113674345265}{134242474613842}a^{8}-\frac{15\!\cdots\!41}{134242474613842}a^{7}+\frac{13\!\cdots\!93}{67121237306921}a^{6}-\frac{23\!\cdots\!55}{134242474613842}a^{5}+\frac{866184727403577}{67121237306921}a^{4}+\frac{17\!\cdots\!05}{134242474613842}a^{3}-\frac{18\!\cdots\!87}{67121237306921}a^{2}+\frac{247272447950517}{19177496373406}a-\frac{5304140008774}{9588748186703}$, $\frac{320499752393}{67121237306921}a^{15}-\frac{211352884502}{9588748186703}a^{14}+\frac{5223266033814}{67121237306921}a^{13}-\frac{19631224740565}{134242474613842}a^{12}+\frac{17174211776145}{134242474613842}a^{11}+\frac{19691090343397}{67121237306921}a^{10}-\frac{142079110730281}{134242474613842}a^{9}+\frac{330333115527705}{134242474613842}a^{8}-\frac{283602627169287}{134242474613842}a^{7}+\frac{175776056325349}{134242474613842}a^{6}+\frac{413350948525357}{134242474613842}a^{5}-\frac{180120262422927}{134242474613842}a^{4}+\frac{102078920534720}{9588748186703}a^{3}+\frac{155247581605678}{67121237306921}a^{2}-\frac{27710363150697}{19177496373406}a+\frac{35344895834830}{9588748186703}$, $\frac{4108689096051}{469848661148447}a^{15}-\frac{62306169219299}{939697322296894}a^{14}+\frac{146756491302229}{469848661148447}a^{13}-\frac{477851566284643}{469848661148447}a^{12}+\frac{11\!\cdots\!09}{469848661148447}a^{11}-\frac{20\!\cdots\!14}{469848661148447}a^{10}+\frac{56\!\cdots\!53}{939697322296894}a^{9}-\frac{47\!\cdots\!51}{939697322296894}a^{8}+\frac{914293023568081}{939697322296894}a^{7}+\frac{50\!\cdots\!71}{939697322296894}a^{6}-\frac{43\!\cdots\!33}{469848661148447}a^{5}+\frac{12\!\cdots\!89}{939697322296894}a^{4}-\frac{64\!\cdots\!29}{939697322296894}a^{3}-\frac{17\!\cdots\!33}{939697322296894}a^{2}+\frac{234072887703729}{134242474613842}a+\frac{34641949374602}{67121237306921}$, $\frac{8407771261545}{939697322296894}a^{15}-\frac{51877016171143}{939697322296894}a^{14}+\frac{106210669244459}{469848661148447}a^{13}-\frac{567878546814513}{939697322296894}a^{12}+\frac{10\!\cdots\!99}{939697322296894}a^{11}-\frac{10\!\cdots\!71}{939697322296894}a^{10}-\frac{8240223096733}{939697322296894}a^{9}+\frac{32\!\cdots\!57}{939697322296894}a^{8}-\frac{63\!\cdots\!49}{939697322296894}a^{7}+\frac{39\!\cdots\!06}{469848661148447}a^{6}-\frac{17\!\cdots\!04}{469848661148447}a^{5}+\frac{13\!\cdots\!41}{939697322296894}a^{4}+\frac{58\!\cdots\!74}{469848661148447}a^{3}-\frac{59\!\cdots\!48}{469848661148447}a^{2}+\frac{150284715108981}{67121237306921}a-\frac{116948412033645}{67121237306921}$, $\frac{6347859944439}{939697322296894}a^{15}-\frac{18419250908132}{469848661148447}a^{14}+\frac{153318615894741}{939697322296894}a^{13}-\frac{208428236935682}{469848661148447}a^{12}+\frac{840121273286595}{939697322296894}a^{11}-\frac{562783104178639}{469848661148447}a^{10}+\frac{10\!\cdots\!05}{939697322296894}a^{9}+\frac{304577534977659}{939697322296894}a^{8}-\frac{13\!\cdots\!21}{939697322296894}a^{7}+\frac{14\!\cdots\!80}{469848661148447}a^{6}-\frac{14\!\cdots\!71}{939697322296894}a^{5}+\frac{22\!\cdots\!18}{469848661148447}a^{4}+\frac{45\!\cdots\!77}{939697322296894}a^{3}+\frac{11\!\cdots\!63}{469848661148447}a^{2}-\frac{39521715782607}{134242474613842}a+\frac{10729834970500}{67121237306921}$, $\frac{782519906424}{67121237306921}a^{15}-\frac{630129249313}{9588748186703}a^{14}+\frac{35884726506765}{134242474613842}a^{13}-\frac{46743913428007}{67121237306921}a^{12}+\frac{176352569170153}{134242474613842}a^{11}-\frac{199006769855261}{134242474613842}a^{10}+\frac{54775411190105}{67121237306921}a^{9}+\frac{149933264620074}{67121237306921}a^{8}-\frac{275526312086799}{67121237306921}a^{7}+\frac{419335957810530}{67121237306921}a^{6}-\frac{12103469928293}{9588748186703}a^{5}+\frac{669828559458137}{134242474613842}a^{4}+\frac{960724381126957}{67121237306921}a^{3}+\frac{42762924619295}{134242474613842}a^{2}+\frac{59463571551711}{19177496373406}a+\frac{9581715934117}{9588748186703}$
| 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: | \( 244244.4706544247 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 244244.4706544247 \cdot 2}{2\cdot\sqrt{202428293557942478212641}}\cr\approx \mathstrut & 1.31864557128249 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 31*x^14 - 93*x^13 + 202*x^12 - 300*x^11 + 277*x^10 + 65*x^9 - 605*x^8 + 1100*x^7 - 946*x^6 + 740*x^5 + 640*x^4 - 1463*x^3 + 715*x^2 - 42*x + 28)
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 - 7*x^15 + 31*x^14 - 93*x^13 + 202*x^12 - 300*x^11 + 277*x^10 + 65*x^9 - 605*x^8 + 1100*x^7 - 946*x^6 + 740*x^5 + 640*x^4 - 1463*x^3 + 715*x^2 - 42*x + 28, 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 - 7*x^15 + 31*x^14 - 93*x^13 + 202*x^12 - 300*x^11 + 277*x^10 + 65*x^9 - 605*x^8 + 1100*x^7 - 946*x^6 + 740*x^5 + 640*x^4 - 1463*x^3 + 715*x^2 - 42*x + 28);
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 - 7*x^15 + 31*x^14 - 93*x^13 + 202*x^12 - 300*x^11 + 277*x^10 + 65*x^9 - 605*x^8 + 1100*x^7 - 946*x^6 + 740*x^5 + 640*x^4 - 1463*x^3 + 715*x^2 - 42*x + 28);
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
| \(\Q(\sqrt{-7}) \), 4.0.670761.1, 8.0.449920319121.4 |
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 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 | ${\href{/padicField/2.6.0.1}{6} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 3.12.16.40 | $x^{12} + 24 x^{11} + 216 x^{10} + 840 x^{9} + 864 x^{8} - 2592 x^{7} - 4482 x^{6} + 8424 x^{5} + 25272 x^{4} + 4968 x^{3} + 29808 x^{2} + 139077$ | $3$ | $4$ | $16$ | $C_{12}$ | $[2]^{4}$ | |
|
\(7\)
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(13\)
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 13.12.8.3 | $x^{12} - 78 x^{9} + 2197 x^{6} + 290004 x^{3} + 114244$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.117.3t1.a.a | $1$ | $ 3^{2} \cdot 13 $ | 3.3.13689.2 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.819.6t1.f.a | $1$ | $ 3^{2} \cdot 7 \cdot 13 $ | 6.0.64274331303.6 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.117.3t1.a.b | $1$ | $ 3^{2} \cdot 13 $ | 3.3.13689.2 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.819.6t1.f.b | $1$ | $ 3^{2} \cdot 7 \cdot 13 $ | 6.0.64274331303.6 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 2.95823.24t21.a.a | $2$ | $ 3^{4} \cdot 7 \cdot 13^{2}$ | 16.0.202428293557942478212641.1 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ | |
| 2.95823.24t21.a.b | $2$ | $ 3^{4} \cdot 7 \cdot 13^{2}$ | 16.0.202428293557942478212641.1 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ | |
| * | 2.819.16t60.a.a | $2$ | $ 3^{2} \cdot 7 \cdot 13 $ | 16.0.202428293557942478212641.1 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ |
| * | 2.819.16t60.a.b | $2$ | $ 3^{2} \cdot 7 \cdot 13 $ | 16.0.202428293557942478212641.1 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ |
| * | 2.819.16t60.a.c | $2$ | $ 3^{2} \cdot 7 \cdot 13 $ | 16.0.202428293557942478212641.1 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ |
| * | 2.819.16t60.a.d | $2$ | $ 3^{2} \cdot 7 \cdot 13 $ | 16.0.202428293557942478212641.1 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ |
| * | 3.95823.6t6.a.a | $3$ | $ 3^{4} \cdot 7 \cdot 13^{2}$ | 6.4.1311721047.2 | $A_4\times C_2$ (as 6T6) | $1$ | $1$ |
| * | 3.670761.4t4.a.a | $3$ | $ 3^{4} \cdot 7^{2} \cdot 13^{2}$ | 4.0.670761.1 | $A_4$ (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 *.