Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 + 8*x^12 - 20*x^11 + 103*x^10 - 127*x^9 + 309*x^8 - 587*x^7 + 1941*x^6 - 586*x^5 + 2442*x^4 - 381*x^3 + 897*x^2 - 67*x + 101)
gp: K = bnfinit(y^14 - 3*y^13 + 8*y^12 - 20*y^11 + 103*y^10 - 127*y^9 + 309*y^8 - 587*y^7 + 1941*y^6 - 586*y^5 + 2442*y^4 - 381*y^3 + 897*y^2 - 67*y + 101, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 3*x^13 + 8*x^12 - 20*x^11 + 103*x^10 - 127*x^9 + 309*x^8 - 587*x^7 + 1941*x^6 - 586*x^5 + 2442*x^4 - 381*x^3 + 897*x^2 - 67*x + 101);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 3*x^13 + 8*x^12 - 20*x^11 + 103*x^10 - 127*x^9 + 309*x^8 - 587*x^7 + 1941*x^6 - 586*x^5 + 2442*x^4 - 381*x^3 + 897*x^2 - 67*x + 101)
\( x^{14} - 3 x^{13} + 8 x^{12} - 20 x^{11} + 103 x^{10} - 127 x^{9} + 309 x^{8} - 587 x^{7} + 1941 x^{6} + \cdots + 101 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-14286214872311167290983\)
\(\medspace = -\,23^{7}\cdot 127^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(38.24\) | 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: | $23^{1/2}127^{6/7}\approx 304.8767107346861$ | ||
| Ramified primes: |
\(23\), \(127\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{103}a^{12}-\frac{30}{103}a^{11}+\frac{43}{103}a^{10}+\frac{27}{103}a^{9}+\frac{39}{103}a^{8}+\frac{40}{103}a^{7}+\frac{7}{103}a^{6}+\frac{51}{103}a^{5}-\frac{20}{103}a^{4}-\frac{19}{103}a^{3}+\frac{18}{103}a^{2}-\frac{47}{103}a-\frac{42}{103}$, $\frac{1}{29\!\cdots\!53}a^{13}+\frac{934952402869162}{29\!\cdots\!53}a^{12}-\frac{86\!\cdots\!36}{29\!\cdots\!53}a^{11}+\frac{14\!\cdots\!48}{29\!\cdots\!53}a^{10}+\frac{13\!\cdots\!69}{29\!\cdots\!53}a^{9}-\frac{23\!\cdots\!90}{29\!\cdots\!53}a^{8}-\frac{54\!\cdots\!69}{29\!\cdots\!53}a^{7}-\frac{73\!\cdots\!78}{29\!\cdots\!53}a^{6}+\frac{30\!\cdots\!54}{29\!\cdots\!53}a^{5}+\frac{43\!\cdots\!07}{29\!\cdots\!53}a^{4}-\frac{94\!\cdots\!36}{29\!\cdots\!53}a^{3}-\frac{10\!\cdots\!58}{29\!\cdots\!53}a^{2}+\frac{12\!\cdots\!45}{29\!\cdots\!53}a-\frac{40\!\cdots\!33}{29\!\cdots\!53}$
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
$C_{3}$, 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: | $6$ | 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: |
$\frac{19\!\cdots\!08}{29\!\cdots\!53}a^{13}-\frac{53\!\cdots\!87}{29\!\cdots\!53}a^{12}+\frac{13\!\cdots\!73}{29\!\cdots\!53}a^{11}-\frac{34\!\cdots\!05}{29\!\cdots\!53}a^{10}+\frac{18\!\cdots\!75}{29\!\cdots\!53}a^{9}-\frac{20\!\cdots\!22}{29\!\cdots\!53}a^{8}+\frac{49\!\cdots\!95}{29\!\cdots\!53}a^{7}-\frac{96\!\cdots\!49}{29\!\cdots\!53}a^{6}+\frac{33\!\cdots\!59}{29\!\cdots\!53}a^{5}-\frac{24\!\cdots\!75}{29\!\cdots\!53}a^{4}+\frac{35\!\cdots\!44}{29\!\cdots\!53}a^{3}+\frac{51\!\cdots\!91}{29\!\cdots\!53}a^{2}+\frac{58\!\cdots\!92}{29\!\cdots\!53}a+\frac{86\!\cdots\!03}{29\!\cdots\!53}$, $\frac{19\!\cdots\!23}{29\!\cdots\!53}a^{13}-\frac{81\!\cdots\!94}{29\!\cdots\!53}a^{12}+\frac{21\!\cdots\!22}{29\!\cdots\!53}a^{11}-\frac{55\!\cdots\!78}{29\!\cdots\!53}a^{10}+\frac{24\!\cdots\!13}{29\!\cdots\!53}a^{9}-\frac{47\!\cdots\!13}{29\!\cdots\!53}a^{8}+\frac{83\!\cdots\!15}{29\!\cdots\!53}a^{7}-\frac{17\!\cdots\!90}{29\!\cdots\!53}a^{6}+\frac{49\!\cdots\!15}{29\!\cdots\!53}a^{5}-\frac{52\!\cdots\!64}{29\!\cdots\!53}a^{4}+\frac{49\!\cdots\!68}{29\!\cdots\!53}a^{3}-\frac{50\!\cdots\!38}{29\!\cdots\!53}a^{2}+\frac{15\!\cdots\!14}{29\!\cdots\!53}a-\frac{99\!\cdots\!14}{29\!\cdots\!53}$, $\frac{63\!\cdots\!15}{29\!\cdots\!53}a^{13}-\frac{50\!\cdots\!01}{29\!\cdots\!53}a^{12}+\frac{13\!\cdots\!23}{29\!\cdots\!53}a^{11}-\frac{35\!\cdots\!46}{29\!\cdots\!53}a^{10}+\frac{12\!\cdots\!66}{29\!\cdots\!53}a^{9}-\frac{38\!\cdots\!90}{29\!\cdots\!53}a^{8}+\frac{51\!\cdots\!52}{29\!\cdots\!53}a^{7}-\frac{11\!\cdots\!45}{29\!\cdots\!53}a^{6}+\frac{28\!\cdots\!79}{29\!\cdots\!53}a^{5}-\frac{59\!\cdots\!76}{29\!\cdots\!53}a^{4}+\frac{17\!\cdots\!00}{29\!\cdots\!53}a^{3}-\frac{64\!\cdots\!49}{29\!\cdots\!53}a^{2}+\frac{27\!\cdots\!04}{29\!\cdots\!53}a-\frac{12\!\cdots\!04}{29\!\cdots\!53}$, $\frac{56\!\cdots\!99}{29\!\cdots\!53}a^{13}-\frac{15\!\cdots\!65}{29\!\cdots\!53}a^{12}+\frac{39\!\cdots\!67}{29\!\cdots\!53}a^{11}-\frac{97\!\cdots\!31}{29\!\cdots\!53}a^{10}+\frac{54\!\cdots\!57}{29\!\cdots\!53}a^{9}-\frac{55\!\cdots\!33}{29\!\cdots\!53}a^{8}+\frac{14\!\cdots\!24}{29\!\cdots\!53}a^{7}-\frac{27\!\cdots\!74}{29\!\cdots\!53}a^{6}+\frac{97\!\cdots\!37}{29\!\cdots\!53}a^{5}-\frac{79\!\cdots\!45}{29\!\cdots\!53}a^{4}+\frac{10\!\cdots\!95}{29\!\cdots\!53}a^{3}+\frac{10\!\cdots\!81}{29\!\cdots\!53}a^{2}+\frac{18\!\cdots\!75}{29\!\cdots\!53}a+\frac{54\!\cdots\!11}{29\!\cdots\!53}$, $\frac{541343745185379}{29\!\cdots\!53}a^{13}-\frac{33\!\cdots\!87}{29\!\cdots\!53}a^{12}+\frac{90\!\cdots\!41}{29\!\cdots\!53}a^{11}-\frac{23\!\cdots\!48}{29\!\cdots\!53}a^{10}+\frac{86\!\cdots\!41}{29\!\cdots\!53}a^{9}-\frac{23\!\cdots\!89}{29\!\cdots\!53}a^{8}+\frac{34\!\cdots\!40}{29\!\cdots\!53}a^{7}-\frac{79\!\cdots\!81}{29\!\cdots\!53}a^{6}+\frac{19\!\cdots\!74}{29\!\cdots\!53}a^{5}-\frac{34\!\cdots\!43}{29\!\cdots\!53}a^{4}+\frac{14\!\cdots\!94}{29\!\cdots\!53}a^{3}-\frac{39\!\cdots\!18}{29\!\cdots\!53}a^{2}+\frac{11\!\cdots\!12}{29\!\cdots\!53}a-\frac{67\!\cdots\!73}{29\!\cdots\!53}$, $\frac{30\!\cdots\!40}{29\!\cdots\!53}a^{13}-\frac{10\!\cdots\!15}{29\!\cdots\!53}a^{12}+\frac{28\!\cdots\!48}{29\!\cdots\!53}a^{11}-\frac{68\!\cdots\!46}{29\!\cdots\!53}a^{10}+\frac{33\!\cdots\!42}{29\!\cdots\!53}a^{9}-\frac{51\!\cdots\!15}{29\!\cdots\!53}a^{8}+\frac{10\!\cdots\!40}{29\!\cdots\!53}a^{7}-\frac{19\!\cdots\!54}{29\!\cdots\!53}a^{6}+\frac{63\!\cdots\!63}{29\!\cdots\!53}a^{5}-\frac{37\!\cdots\!16}{29\!\cdots\!53}a^{4}+\frac{57\!\cdots\!06}{29\!\cdots\!53}a^{3}-\frac{82\!\cdots\!69}{29\!\cdots\!53}a^{2}+\frac{10\!\cdots\!39}{29\!\cdots\!53}a+\frac{80\!\cdots\!26}{29\!\cdots\!53}$
| 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: | \( 57111.48044975551 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 57111.48044975551 \cdot 3}{2\cdot\sqrt{14286214872311167290983}}\cr\approx \mathstrut & 0.277086372296149 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 + 8*x^12 - 20*x^11 + 103*x^10 - 127*x^9 + 309*x^8 - 587*x^7 + 1941*x^6 - 586*x^5 + 2442*x^4 - 381*x^3 + 897*x^2 - 67*x + 101)
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^14 - 3*x^13 + 8*x^12 - 20*x^11 + 103*x^10 - 127*x^9 + 309*x^8 - 587*x^7 + 1941*x^6 - 586*x^5 + 2442*x^4 - 381*x^3 + 897*x^2 - 67*x + 101, 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^14 - 3*x^13 + 8*x^12 - 20*x^11 + 103*x^10 - 127*x^9 + 309*x^8 - 587*x^7 + 1941*x^6 - 586*x^5 + 2442*x^4 - 381*x^3 + 897*x^2 - 67*x + 101);
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^14 - 3*x^13 + 8*x^12 - 20*x^11 + 103*x^10 - 127*x^9 + 309*x^8 - 587*x^7 + 1941*x^6 - 586*x^5 + 2442*x^4 - 381*x^3 + 897*x^2 - 67*x + 101);
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
$C_7\times D_7$ (as 14T8):
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 98 |
| The 35 conjugacy class representatives for $C_7 \wr C_2$ |
| Character table for $C_7 \wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-23}) \) |
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 14 siblings: | deg 14, deg 14 |
| 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 | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.14.0.1}{14} }$ | ${\href{/padicField/11.14.0.1}{14} }$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | ${\href{/padicField/17.14.0.1}{14} }$ | ${\href{/padicField/19.2.0.1}{2} }^{7}$ | R | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{7}$ | ${\href{/padicField/31.7.0.1}{7} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{7}$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{7}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{7}$ | ${\href{/padicField/53.14.0.1}{14} }$ | ${\href{/padicField/59.7.0.1}{7} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.14.7.2 | $x^{14} - 21574 x^{13} + 234933293 x^{12} - 1215548742590 x^{11} + 4112603302919993 x^{10} - 2725136947640868418 x^{9} + 363970304488058959670 x^{8} + 412439955621146008597774 x^{7} + 8371317003225356072410 x^{6} - 1441597445302019393122 x^{5} + 50038044386627554831 x^{4} - 340160375675128190 x^{3} + 1512111255867499 x^{2} - 3193726269286 x + 3404825447$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ |
|
\(127\)
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 127.7.6.1 | $x^{7} + 127$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.23.2t1.a.a | $1$ | $ 23 $ | \(\Q(\sqrt{-23}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.2921.14t1.a.f | $1$ | $ 23 \cdot 127 $ | 14.0.59943142036157597462980720397949287.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.127.7t1.a.f | $1$ | $ 127 $ | 7.7.4195872914689.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.127.7t1.a.e | $1$ | $ 127 $ | 7.7.4195872914689.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.2921.14t1.a.a | $1$ | $ 23 \cdot 127 $ | 14.0.59943142036157597462980720397949287.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.2921.14t1.a.d | $1$ | $ 23 \cdot 127 $ | 14.0.59943142036157597462980720397949287.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.2921.14t1.a.c | $1$ | $ 23 \cdot 127 $ | 14.0.59943142036157597462980720397949287.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.127.7t1.a.b | $1$ | $ 127 $ | 7.7.4195872914689.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.127.7t1.a.d | $1$ | $ 127 $ | 7.7.4195872914689.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.127.7t1.a.a | $1$ | $ 127 $ | 7.7.4195872914689.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.127.7t1.a.c | $1$ | $ 127 $ | 7.7.4195872914689.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.2921.14t1.a.e | $1$ | $ 23 \cdot 127 $ | 14.0.59943142036157597462980720397949287.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.2921.14t1.a.b | $1$ | $ 23 \cdot 127 $ | 14.0.59943142036157597462980720397949287.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| * | 2.2921.14t8.a.b | $2$ | $ 23 \cdot 127 $ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| * | 2.2921.14t8.a.a | $2$ | $ 23 \cdot 127 $ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.370967.14t8.a.f | $2$ | $ 23 \cdot 127^{2}$ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| * | 2.2921.14t8.a.e | $2$ | $ 23 \cdot 127 $ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.370967.14t8.a.d | $2$ | $ 23 \cdot 127^{2}$ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.370967.7t2.a.a | $2$ | $ 23 \cdot 127^{2}$ | 7.1.51051185753021063.1 | $D_{7}$ (as 7T2) | $1$ | $0$ | |
| 2.370967.7t2.a.c | $2$ | $ 23 \cdot 127^{2}$ | 7.1.51051185753021063.1 | $D_{7}$ (as 7T2) | $1$ | $0$ | |
| 2.370967.14t8.b.c | $2$ | $ 23 \cdot 127^{2}$ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.370967.14t8.b.a | $2$ | $ 23 \cdot 127^{2}$ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.370967.14t8.a.a | $2$ | $ 23 \cdot 127^{2}$ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| * | 2.2921.14t8.a.f | $2$ | $ 23 \cdot 127 $ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.370967.14t8.b.f | $2$ | $ 23 \cdot 127^{2}$ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| * | 2.2921.14t8.a.d | $2$ | $ 23 \cdot 127 $ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.370967.7t2.a.b | $2$ | $ 23 \cdot 127^{2}$ | 7.1.51051185753021063.1 | $D_{7}$ (as 7T2) | $1$ | $0$ | |
| 2.370967.14t8.b.b | $2$ | $ 23 \cdot 127^{2}$ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.370967.14t8.b.e | $2$ | $ 23 \cdot 127^{2}$ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.370967.14t8.a.e | $2$ | $ 23 \cdot 127^{2}$ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| * | 2.2921.14t8.a.c | $2$ | $ 23 \cdot 127 $ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.370967.14t8.a.b | $2$ | $ 23 \cdot 127^{2}$ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.370967.14t8.b.d | $2$ | $ 23 \cdot 127^{2}$ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.370967.14t8.a.c | $2$ | $ 23 \cdot 127^{2}$ | 14.0.14286214872311167290983.1 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
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 *.