Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 188*x^10 - 213*x^9 + 12715*x^8 + 39298*x^7 - 324717*x^6 - 1733319*x^5 + 680764*x^4 + 20357101*x^3 + 48082157*x^2 + 40734254*x + 8274052)
gp: K = bnfinit(y^12 - y^11 - 188*y^10 - 213*y^9 + 12715*y^8 + 39298*y^7 - 324717*y^6 - 1733319*y^5 + 680764*y^4 + 20357101*y^3 + 48082157*y^2 + 40734254*y + 8274052, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 188*x^10 - 213*x^9 + 12715*x^8 + 39298*x^7 - 324717*x^6 - 1733319*x^5 + 680764*x^4 + 20357101*x^3 + 48082157*x^2 + 40734254*x + 8274052);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 188*x^10 - 213*x^9 + 12715*x^8 + 39298*x^7 - 324717*x^6 - 1733319*x^5 + 680764*x^4 + 20357101*x^3 + 48082157*x^2 + 40734254*x + 8274052)
\( x^{12} - x^{11} - 188 x^{10} - 213 x^{9} + 12715 x^{8} + 39298 x^{7} - 324717 x^{6} - 1733319 x^{5} + \cdots + 8274052 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(8025462320079492591473139857\)
\(\medspace = 17^{9}\cdot 127^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(211.53\) | 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: | $17^{3/4}127^{2/3}\approx 211.53026087633702$ | ||
| Ramified primes: |
\(17\), \(127\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2159=17\cdot 127\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2159}(1,·)$, $\chi_{2159}(234,·)$, $\chi_{2159}(273,·)$, $\chi_{2159}(361,·)$, $\chi_{2159}(509,·)$, $\chi_{2159}(781,·)$, $\chi_{2159}(1123,·)$, $\chi_{2159}(1271,·)$, $\chi_{2159}(1398,·)$, $\chi_{2159}(1543,·)$, $\chi_{2159}(1631,·)$, $\chi_{2159}(1670,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{4}+\frac{1}{8}a^{3}-\frac{3}{8}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{62\!\cdots\!44}a^{11}-\frac{25\!\cdots\!13}{31\!\cdots\!72}a^{10}+\frac{14\!\cdots\!63}{31\!\cdots\!72}a^{9}-\frac{47\!\cdots\!71}{62\!\cdots\!44}a^{8}-\frac{99\!\cdots\!27}{31\!\cdots\!72}a^{7}+\frac{15\!\cdots\!12}{38\!\cdots\!09}a^{6}+\frac{29\!\cdots\!03}{62\!\cdots\!44}a^{5}-\frac{55\!\cdots\!85}{31\!\cdots\!72}a^{4}+\frac{58\!\cdots\!19}{31\!\cdots\!72}a^{3}+\frac{18\!\cdots\!03}{62\!\cdots\!44}a^{2}-\frac{57\!\cdots\!35}{31\!\cdots\!72}a-\frac{60\!\cdots\!23}{15\!\cdots\!36}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{3}$, which has order $3$ (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: |
\( -1 \)
(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: |
$\frac{98\!\cdots\!99}{62\!\cdots\!44}a^{11}-\frac{20\!\cdots\!73}{31\!\cdots\!72}a^{10}-\frac{85\!\cdots\!51}{31\!\cdots\!72}a^{9}+\frac{31\!\cdots\!63}{62\!\cdots\!44}a^{8}+\frac{56\!\cdots\!11}{31\!\cdots\!72}a^{7}+\frac{47\!\cdots\!15}{77\!\cdots\!18}a^{6}-\frac{32\!\cdots\!47}{62\!\cdots\!44}a^{5}-\frac{35\!\cdots\!45}{31\!\cdots\!72}a^{4}+\frac{13\!\cdots\!45}{31\!\cdots\!72}a^{3}+\frac{11\!\cdots\!85}{62\!\cdots\!44}a^{2}+\frac{63\!\cdots\!39}{31\!\cdots\!72}a+\frac{70\!\cdots\!43}{15\!\cdots\!36}$, $\frac{34\!\cdots\!57}{31\!\cdots\!72}a^{11}-\frac{28\!\cdots\!57}{77\!\cdots\!18}a^{10}-\frac{15\!\cdots\!09}{77\!\cdots\!18}a^{9}+\frac{71\!\cdots\!29}{31\!\cdots\!72}a^{8}+\frac{21\!\cdots\!23}{15\!\cdots\!36}a^{7}+\frac{46\!\cdots\!25}{38\!\cdots\!09}a^{6}-\frac{12\!\cdots\!53}{31\!\cdots\!72}a^{5}-\frac{79\!\cdots\!15}{77\!\cdots\!18}a^{4}+\frac{12\!\cdots\!00}{38\!\cdots\!09}a^{3}+\frac{47\!\cdots\!75}{31\!\cdots\!72}a^{2}+\frac{27\!\cdots\!13}{15\!\cdots\!36}a+\frac{32\!\cdots\!19}{77\!\cdots\!18}$, $\frac{16\!\cdots\!13}{62\!\cdots\!44}a^{11}-\frac{31\!\cdots\!01}{31\!\cdots\!72}a^{10}-\frac{14\!\cdots\!87}{31\!\cdots\!72}a^{9}+\frac{46\!\cdots\!21}{62\!\cdots\!44}a^{8}+\frac{99\!\cdots\!57}{31\!\cdots\!72}a^{7}+\frac{14\!\cdots\!65}{77\!\cdots\!18}a^{6}-\frac{56\!\cdots\!53}{62\!\cdots\!44}a^{5}-\frac{67\!\cdots\!05}{31\!\cdots\!72}a^{4}+\frac{23\!\cdots\!45}{31\!\cdots\!72}a^{3}+\frac{21\!\cdots\!35}{62\!\cdots\!44}a^{2}+\frac{11\!\cdots\!65}{31\!\cdots\!72}a+\frac{13\!\cdots\!81}{15\!\cdots\!36}$, $\frac{11\!\cdots\!37}{62\!\cdots\!44}a^{11}+\frac{57\!\cdots\!03}{38\!\cdots\!09}a^{10}-\frac{71\!\cdots\!95}{15\!\cdots\!36}a^{9}-\frac{17\!\cdots\!77}{62\!\cdots\!44}a^{8}+\frac{11\!\cdots\!35}{31\!\cdots\!72}a^{7}+\frac{84\!\cdots\!88}{38\!\cdots\!09}a^{6}-\frac{62\!\cdots\!93}{62\!\cdots\!44}a^{5}-\frac{11\!\cdots\!07}{15\!\cdots\!36}a^{4}+\frac{24\!\cdots\!49}{77\!\cdots\!18}a^{3}+\frac{47\!\cdots\!97}{62\!\cdots\!44}a^{2}+\frac{43\!\cdots\!63}{31\!\cdots\!72}a+\frac{10\!\cdots\!67}{15\!\cdots\!36}$, $\frac{29\!\cdots\!31}{31\!\cdots\!72}a^{11}-\frac{42\!\cdots\!59}{31\!\cdots\!72}a^{10}-\frac{60\!\cdots\!55}{31\!\cdots\!72}a^{9}-\frac{42\!\cdots\!79}{15\!\cdots\!36}a^{8}+\frac{54\!\cdots\!12}{38\!\cdots\!09}a^{7}+\frac{30\!\cdots\!07}{77\!\cdots\!18}a^{6}-\frac{12\!\cdots\!67}{31\!\cdots\!72}a^{5}-\frac{52\!\cdots\!61}{31\!\cdots\!72}a^{4}+\frac{84\!\cdots\!07}{31\!\cdots\!72}a^{3}+\frac{33\!\cdots\!25}{15\!\cdots\!36}a^{2}+\frac{11\!\cdots\!59}{38\!\cdots\!09}a+\frac{26\!\cdots\!50}{38\!\cdots\!09}$, $\frac{11\!\cdots\!63}{15\!\cdots\!36}a^{11}-\frac{30\!\cdots\!05}{15\!\cdots\!36}a^{10}-\frac{53\!\cdots\!93}{38\!\cdots\!09}a^{9}+\frac{56\!\cdots\!25}{77\!\cdots\!18}a^{8}+\frac{73\!\cdots\!49}{77\!\cdots\!18}a^{7}+\frac{10\!\cdots\!31}{77\!\cdots\!18}a^{6}-\frac{41\!\cdots\!43}{15\!\cdots\!36}a^{5}-\frac{12\!\cdots\!15}{15\!\cdots\!36}a^{4}+\frac{15\!\cdots\!43}{77\!\cdots\!18}a^{3}+\frac{91\!\cdots\!15}{77\!\cdots\!18}a^{2}+\frac{11\!\cdots\!67}{77\!\cdots\!18}a+\frac{12\!\cdots\!68}{38\!\cdots\!09}$, $\frac{56\!\cdots\!25}{31\!\cdots\!72}a^{11}-\frac{38\!\cdots\!11}{15\!\cdots\!36}a^{10}-\frac{55\!\cdots\!01}{15\!\cdots\!36}a^{9}-\frac{51\!\cdots\!51}{31\!\cdots\!72}a^{8}+\frac{39\!\cdots\!55}{15\!\cdots\!36}a^{7}+\frac{20\!\cdots\!74}{38\!\cdots\!09}a^{6}-\frac{22\!\cdots\!73}{31\!\cdots\!72}a^{5}-\frac{39\!\cdots\!43}{15\!\cdots\!36}a^{4}+\frac{81\!\cdots\!39}{15\!\cdots\!36}a^{3}+\frac{10\!\cdots\!99}{31\!\cdots\!72}a^{2}+\frac{67\!\cdots\!23}{15\!\cdots\!36}a+\frac{77\!\cdots\!01}{77\!\cdots\!18}$, $\frac{73\!\cdots\!13}{62\!\cdots\!44}a^{11}-\frac{14\!\cdots\!51}{31\!\cdots\!72}a^{10}-\frac{65\!\cdots\!73}{31\!\cdots\!72}a^{9}+\frac{21\!\cdots\!61}{62\!\cdots\!44}a^{8}+\frac{43\!\cdots\!93}{31\!\cdots\!72}a^{7}+\frac{25\!\cdots\!35}{38\!\cdots\!09}a^{6}-\frac{25\!\cdots\!05}{62\!\cdots\!44}a^{5}-\frac{28\!\cdots\!35}{31\!\cdots\!72}a^{4}+\frac{10\!\cdots\!31}{31\!\cdots\!72}a^{3}+\frac{89\!\cdots\!35}{62\!\cdots\!44}a^{2}+\frac{48\!\cdots\!05}{31\!\cdots\!72}a+\frac{53\!\cdots\!85}{15\!\cdots\!36}$, $\frac{48\!\cdots\!75}{15\!\cdots\!36}a^{11}-\frac{29\!\cdots\!87}{31\!\cdots\!72}a^{10}-\frac{17\!\cdots\!25}{31\!\cdots\!72}a^{9}+\frac{14\!\cdots\!93}{31\!\cdots\!72}a^{8}+\frac{60\!\cdots\!47}{15\!\cdots\!36}a^{7}+\frac{17\!\cdots\!26}{38\!\cdots\!09}a^{6}-\frac{17\!\cdots\!03}{15\!\cdots\!36}a^{5}-\frac{10\!\cdots\!61}{31\!\cdots\!72}a^{4}+\frac{26\!\cdots\!09}{31\!\cdots\!72}a^{3}+\frac{14\!\cdots\!83}{31\!\cdots\!72}a^{2}+\frac{89\!\cdots\!29}{15\!\cdots\!36}a+\frac{10\!\cdots\!01}{77\!\cdots\!18}$, $\frac{98\!\cdots\!13}{31\!\cdots\!72}a^{11}-\frac{10\!\cdots\!51}{15\!\cdots\!36}a^{10}-\frac{94\!\cdots\!37}{15\!\cdots\!36}a^{9}+\frac{38\!\cdots\!57}{31\!\cdots\!72}a^{8}+\frac{66\!\cdots\!83}{15\!\cdots\!36}a^{7}+\frac{49\!\cdots\!17}{77\!\cdots\!18}a^{6}-\frac{38\!\cdots\!65}{31\!\cdots\!72}a^{5}-\frac{57\!\cdots\!55}{15\!\cdots\!36}a^{4}+\frac{14\!\cdots\!33}{15\!\cdots\!36}a^{3}+\frac{16\!\cdots\!79}{31\!\cdots\!72}a^{2}+\frac{95\!\cdots\!95}{15\!\cdots\!36}a+\frac{10\!\cdots\!39}{77\!\cdots\!18}$, $\frac{18\!\cdots\!03}{15\!\cdots\!36}a^{11}-\frac{14\!\cdots\!93}{31\!\cdots\!72}a^{10}-\frac{64\!\cdots\!41}{31\!\cdots\!72}a^{9}+\frac{10\!\cdots\!07}{31\!\cdots\!72}a^{8}+\frac{21\!\cdots\!29}{15\!\cdots\!36}a^{7}+\frac{24\!\cdots\!30}{38\!\cdots\!09}a^{6}-\frac{62\!\cdots\!25}{15\!\cdots\!36}a^{5}-\frac{28\!\cdots\!71}{31\!\cdots\!72}a^{4}+\frac{10\!\cdots\!65}{31\!\cdots\!72}a^{3}+\frac{44\!\cdots\!29}{31\!\cdots\!72}a^{2}+\frac{24\!\cdots\!21}{15\!\cdots\!36}a+\frac{26\!\cdots\!11}{77\!\cdots\!18}$
| 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: | \( 13717390098.347887 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 13717390098.347887 \cdot 3}{2\cdot\sqrt{8025462320079492591473139857}}\cr\approx \mathstrut & 0.940779112269357 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 188*x^10 - 213*x^9 + 12715*x^8 + 39298*x^7 - 324717*x^6 - 1733319*x^5 + 680764*x^4 + 20357101*x^3 + 48082157*x^2 + 40734254*x + 8274052)
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^12 - x^11 - 188*x^10 - 213*x^9 + 12715*x^8 + 39298*x^7 - 324717*x^6 - 1733319*x^5 + 680764*x^4 + 20357101*x^3 + 48082157*x^2 + 40734254*x + 8274052, 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^12 - x^11 - 188*x^10 - 213*x^9 + 12715*x^8 + 39298*x^7 - 324717*x^6 - 1733319*x^5 + 680764*x^4 + 20357101*x^3 + 48082157*x^2 + 40734254*x + 8274052);
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^12 - x^11 - 188*x^10 - 213*x^9 + 12715*x^8 + 39298*x^7 - 324717*x^6 - 1733319*x^5 + 680764*x^4 + 20357101*x^3 + 48082157*x^2 + 40734254*x + 8274052);
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
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 cyclic group of order 12 |
| The 12 conjugacy class representatives for $C_{12}$ |
| Character table for $C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 3.3.16129.1, 4.4.4913.1, 6.6.1278090621233.3 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{6}$ | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.1.0.1}{1} }^{12}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(127\)
| 127.6.4.1 | $x^{6} + 378 x^{5} + 47637 x^{4} + 2002898 x^{3} + 190917 x^{2} + 6049872 x + 253919890$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 127.6.4.1 | $x^{6} + 378 x^{5} + 47637 x^{4} + 2002898 x^{3} + 190917 x^{2} + 6049872 x + 253919890$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.17.4t1.a.a | $1$ | $ 17 $ | 4.4.4913.1 | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.17.4t1.a.b | $1$ | $ 17 $ | 4.4.4913.1 | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.127.3t1.a.a | $1$ | $ 127 $ | 3.3.16129.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.2159.12t1.a.a | $1$ | $ 17 \cdot 127 $ | 12.12.8025462320079492591473139857.2 | $C_{12}$ (as 12T1) | $0$ | $1$ |
| * | 1.2159.6t1.a.a | $1$ | $ 17 \cdot 127 $ | 6.6.1278090621233.3 | $C_6$ (as 6T1) | $0$ | $1$ |
| * | 1.2159.12t1.a.b | $1$ | $ 17 \cdot 127 $ | 12.12.8025462320079492591473139857.2 | $C_{12}$ (as 12T1) | $0$ | $1$ |
| * | 1.127.3t1.a.b | $1$ | $ 127 $ | 3.3.16129.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.2159.12t1.a.c | $1$ | $ 17 \cdot 127 $ | 12.12.8025462320079492591473139857.2 | $C_{12}$ (as 12T1) | $0$ | $1$ |
| * | 1.2159.6t1.a.b | $1$ | $ 17 \cdot 127 $ | 6.6.1278090621233.3 | $C_6$ (as 6T1) | $0$ | $1$ |
| * | 1.2159.12t1.a.d | $1$ | $ 17 \cdot 127 $ | 12.12.8025462320079492591473139857.2 | $C_{12}$ (as 12T1) | $0$ | $1$ |
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 *.