Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 2*x^12 + 20*x^11 - 83*x^10 + 12*x^9 + 110*x^8 - 176*x^7 + 2447*x^6 - 3952*x^5 - 2038*x^4 - 5862*x^3 + 9737*x^2 + 11592*x + 13033)
gp: K = bnfinit(y^14 - 2*y^13 + 2*y^12 + 20*y^11 - 83*y^10 + 12*y^9 + 110*y^8 - 176*y^7 + 2447*y^6 - 3952*y^5 - 2038*y^4 - 5862*y^3 + 9737*y^2 + 11592*y + 13033, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 2*x^13 + 2*x^12 + 20*x^11 - 83*x^10 + 12*x^9 + 110*x^8 - 176*x^7 + 2447*x^6 - 3952*x^5 - 2038*x^4 - 5862*x^3 + 9737*x^2 + 11592*x + 13033);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 2*x^13 + 2*x^12 + 20*x^11 - 83*x^10 + 12*x^9 + 110*x^8 - 176*x^7 + 2447*x^6 - 3952*x^5 - 2038*x^4 - 5862*x^3 + 9737*x^2 + 11592*x + 13033)
\( x^{14} - 2 x^{13} + 2 x^{12} + 20 x^{11} - 83 x^{10} + 12 x^{9} + 110 x^{8} - 176 x^{7} + 2447 x^{6} + \cdots + 13033 \)
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: |
\(-5796901408038404767744\)
\(\medspace = -\,2^{14}\cdot 29^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(35.85\) | 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: | $2\cdot 29^{6/7}\approx 35.8520500359704$ | ||
| Ramified primes: |
\(2\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{12\!\cdots\!13}a^{13}+\frac{34\!\cdots\!12}{12\!\cdots\!13}a^{12}-\frac{47\!\cdots\!50}{12\!\cdots\!13}a^{11}-\frac{42\!\cdots\!14}{12\!\cdots\!13}a^{10}-\frac{25\!\cdots\!48}{12\!\cdots\!13}a^{9}-\frac{34\!\cdots\!22}{12\!\cdots\!13}a^{8}+\frac{30\!\cdots\!11}{12\!\cdots\!13}a^{7}+\frac{41\!\cdots\!83}{12\!\cdots\!13}a^{6}+\frac{38\!\cdots\!98}{12\!\cdots\!13}a^{5}-\frac{41\!\cdots\!53}{12\!\cdots\!13}a^{4}+\frac{36\!\cdots\!34}{12\!\cdots\!13}a^{3}-\frac{69\!\cdots\!90}{12\!\cdots\!13}a^{2}-\frac{42\!\cdots\!78}{12\!\cdots\!13}a+\frac{52\!\cdots\!24}{12\!\cdots\!13}$
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
$C_{7}$, which has order $7$
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: |
\( \frac{12317228248971}{163719440972021573} a^{13} - \frac{26878028333909}{163719440972021573} a^{12} + \frac{92349634788534}{163719440972021573} a^{11} + \frac{204932961825272}{163719440972021573} a^{10} - \frac{945241267907748}{163719440972021573} a^{9} + \frac{1863823155722795}{163719440972021573} a^{8} - \frac{1854211622966132}{163719440972021573} a^{7} - \frac{4444186303339562}{163719440972021573} a^{6} + \frac{33985587918655108}{163719440972021573} a^{5} - \frac{70680160761132613}{163719440972021573} a^{4} + \frac{120436235845816693}{163719440972021573} a^{3} - \frac{134346294211846567}{163719440972021573} a^{2} - \frac{37553476289553289}{163719440972021573} a - \frac{218804396285973327}{163719440972021573} \)
(order $4$)
| 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: |
$\frac{95\!\cdots\!15}{12\!\cdots\!13}a^{13}+\frac{67\!\cdots\!47}{12\!\cdots\!13}a^{12}-\frac{25\!\cdots\!71}{12\!\cdots\!13}a^{11}+\frac{66\!\cdots\!87}{12\!\cdots\!13}a^{10}+\frac{62\!\cdots\!89}{12\!\cdots\!13}a^{9}-\frac{90\!\cdots\!37}{12\!\cdots\!13}a^{8}+\frac{12\!\cdots\!56}{12\!\cdots\!13}a^{7}+\frac{68\!\cdots\!90}{12\!\cdots\!13}a^{6}-\frac{94\!\cdots\!35}{12\!\cdots\!13}a^{5}+\frac{20\!\cdots\!21}{12\!\cdots\!13}a^{4}-\frac{62\!\cdots\!87}{12\!\cdots\!13}a^{3}+\frac{35\!\cdots\!67}{12\!\cdots\!13}a^{2}-\frac{30\!\cdots\!02}{12\!\cdots\!13}a+\frac{93\!\cdots\!94}{12\!\cdots\!13}$, $\frac{38\!\cdots\!47}{70\!\cdots\!73}a^{13}-\frac{80\!\cdots\!37}{70\!\cdots\!73}a^{12}+\frac{71\!\cdots\!38}{70\!\cdots\!73}a^{11}+\frac{88\!\cdots\!02}{70\!\cdots\!73}a^{10}-\frac{34\!\cdots\!28}{70\!\cdots\!73}a^{9}+\frac{71\!\cdots\!02}{70\!\cdots\!73}a^{8}+\frac{74\!\cdots\!10}{70\!\cdots\!73}a^{7}-\frac{13\!\cdots\!39}{70\!\cdots\!73}a^{6}+\frac{93\!\cdots\!44}{70\!\cdots\!73}a^{5}-\frac{14\!\cdots\!60}{70\!\cdots\!73}a^{4}-\frac{11\!\cdots\!37}{70\!\cdots\!73}a^{3}+\frac{44\!\cdots\!71}{70\!\cdots\!73}a^{2}+\frac{20\!\cdots\!51}{70\!\cdots\!73}a+\frac{31\!\cdots\!33}{70\!\cdots\!73}$, $\frac{21\!\cdots\!38}{12\!\cdots\!13}a^{13}-\frac{99\!\cdots\!44}{12\!\cdots\!13}a^{12}+\frac{16\!\cdots\!27}{12\!\cdots\!13}a^{11}+\frac{38\!\cdots\!66}{12\!\cdots\!13}a^{10}-\frac{31\!\cdots\!56}{12\!\cdots\!13}a^{9}+\frac{53\!\cdots\!81}{12\!\cdots\!13}a^{8}+\frac{29\!\cdots\!13}{12\!\cdots\!13}a^{7}-\frac{19\!\cdots\!25}{12\!\cdots\!13}a^{6}+\frac{70\!\cdots\!29}{12\!\cdots\!13}a^{5}-\frac{20\!\cdots\!02}{12\!\cdots\!13}a^{4}+\frac{14\!\cdots\!77}{12\!\cdots\!13}a^{3}+\frac{20\!\cdots\!88}{12\!\cdots\!13}a^{2}-\frac{91\!\cdots\!77}{12\!\cdots\!13}a-\frac{33\!\cdots\!83}{12\!\cdots\!13}$, $\frac{18\!\cdots\!26}{12\!\cdots\!13}a^{13}-\frac{28\!\cdots\!30}{12\!\cdots\!13}a^{12}+\frac{43\!\cdots\!65}{12\!\cdots\!13}a^{11}+\frac{47\!\cdots\!69}{12\!\cdots\!13}a^{10}-\frac{13\!\cdots\!80}{12\!\cdots\!13}a^{9}-\frac{89\!\cdots\!45}{12\!\cdots\!13}a^{8}+\frac{44\!\cdots\!36}{12\!\cdots\!13}a^{7}-\frac{52\!\cdots\!22}{12\!\cdots\!13}a^{6}+\frac{32\!\cdots\!33}{12\!\cdots\!13}a^{5}-\frac{53\!\cdots\!10}{12\!\cdots\!13}a^{4}-\frac{12\!\cdots\!01}{12\!\cdots\!13}a^{3}+\frac{36\!\cdots\!49}{12\!\cdots\!13}a^{2}+\frac{29\!\cdots\!92}{12\!\cdots\!13}a+\frac{27\!\cdots\!41}{12\!\cdots\!13}$, $\frac{21\!\cdots\!63}{12\!\cdots\!13}a^{13}-\frac{36\!\cdots\!98}{12\!\cdots\!13}a^{12}+\frac{81\!\cdots\!72}{12\!\cdots\!13}a^{11}-\frac{54\!\cdots\!62}{12\!\cdots\!13}a^{10}-\frac{83\!\cdots\!01}{12\!\cdots\!13}a^{9}+\frac{29\!\cdots\!43}{12\!\cdots\!13}a^{8}-\frac{16\!\cdots\!26}{12\!\cdots\!13}a^{7}-\frac{43\!\cdots\!97}{12\!\cdots\!13}a^{6}+\frac{15\!\cdots\!07}{12\!\cdots\!13}a^{5}-\frac{92\!\cdots\!46}{12\!\cdots\!13}a^{4}+\frac{16\!\cdots\!16}{12\!\cdots\!13}a^{3}-\frac{31\!\cdots\!86}{12\!\cdots\!13}a^{2}+\frac{15\!\cdots\!64}{12\!\cdots\!13}a-\frac{30\!\cdots\!11}{12\!\cdots\!13}$, $\frac{11\!\cdots\!87}{12\!\cdots\!13}a^{13}-\frac{22\!\cdots\!16}{12\!\cdots\!13}a^{12}+\frac{15\!\cdots\!24}{12\!\cdots\!13}a^{11}+\frac{31\!\cdots\!68}{12\!\cdots\!13}a^{10}-\frac{10\!\cdots\!12}{12\!\cdots\!13}a^{9}-\frac{34\!\cdots\!42}{12\!\cdots\!13}a^{8}+\frac{40\!\cdots\!46}{12\!\cdots\!13}a^{7}-\frac{50\!\cdots\!25}{12\!\cdots\!13}a^{6}+\frac{22\!\cdots\!84}{12\!\cdots\!13}a^{5}-\frac{26\!\cdots\!10}{12\!\cdots\!13}a^{4}-\frac{77\!\cdots\!53}{12\!\cdots\!13}a^{3}+\frac{26\!\cdots\!48}{12\!\cdots\!13}a^{2}-\frac{47\!\cdots\!04}{12\!\cdots\!13}a+\frac{19\!\cdots\!96}{12\!\cdots\!13}$
| 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: | \( 14379.655158793385 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 14379.655158793385 \cdot 7}{4\cdot\sqrt{5796901408038404767744}}\cr\approx \mathstrut & 0.127775515402617 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 2*x^12 + 20*x^11 - 83*x^10 + 12*x^9 + 110*x^8 - 176*x^7 + 2447*x^6 - 3952*x^5 - 2038*x^4 - 5862*x^3 + 9737*x^2 + 11592*x + 13033)
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 - 2*x^13 + 2*x^12 + 20*x^11 - 83*x^10 + 12*x^9 + 110*x^8 - 176*x^7 + 2447*x^6 - 3952*x^5 - 2038*x^4 - 5862*x^3 + 9737*x^2 + 11592*x + 13033, 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 - 2*x^13 + 2*x^12 + 20*x^11 - 83*x^10 + 12*x^9 + 110*x^8 - 176*x^7 + 2447*x^6 - 3952*x^5 - 2038*x^4 - 5862*x^3 + 9737*x^2 + 11592*x + 13033);
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 - 2*x^13 + 2*x^12 + 20*x^11 - 83*x^10 + 12*x^9 + 110*x^8 - 176*x^7 + 2447*x^6 - 3952*x^5 - 2038*x^4 - 5862*x^3 + 9737*x^2 + 11592*x + 13033);
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
| \(\Q(\sqrt{-1}) \) |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.14.0.1}{14} }$ | ${\href{/padicField/11.14.0.1}{14} }$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | ${\href{/padicField/17.7.0.1}{7} }^{2}$ | ${\href{/padicField/19.14.0.1}{14} }$ | ${\href{/padicField/23.14.0.1}{14} }$ | R | ${\href{/padicField/31.14.0.1}{14} }$ | ${\href{/padicField/37.7.0.1}{7} }^{2}$ | ${\href{/padicField/41.7.0.1}{7} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.14.0.1}{14} }$ | ${\href{/padicField/53.7.0.1}{7} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.14.14.38 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504$ | $2$ | $7$ | $14$ | $C_{14}$ | $[2]^{7}$ |
|
\(29\)
| 29.7.6.5 | $x^{7} + 58$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
| 29.7.6.6 | $x^{7} + 319$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.116.14t1.b.c | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.29.7t1.a.e | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.116.14t1.b.d | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.29.7t1.a.f | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.116.14t1.b.e | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.116.14t1.b.b | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.116.14t1.b.a | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.29.7t1.a.b | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.116.14t1.b.f | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.29.7t1.a.c | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.29.7t1.a.d | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.29.7t1.a.a | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 2.3364.7t2.a.c | $2$ | $ 2^{2} \cdot 29^{2}$ | 7.1.38068692544.1 | $D_{7}$ (as 7T2) | $1$ | $0$ | |
| 2.3364.14t8.a.c | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.3364.14t8.a.b | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.3364.14t8.a.e | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.116.14t8.a.d | $2$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.116.14t8.a.f | $2$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.3364.14t8.a.a | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.3364.7t2.a.b | $2$ | $ 2^{2} \cdot 29^{2}$ | 7.1.38068692544.1 | $D_{7}$ (as 7T2) | $1$ | $0$ | |
| 2.3364.14t8.a.f | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.116.14t8.a.e | $2$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.116.14t8.a.a | $2$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| * | 2.3364.14t8.b.e | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.116.14t8.a.b | $2$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.116.14t8.a.c | $2$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| * | 2.3364.14t8.b.b | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| * | 2.3364.14t8.b.c | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.3364.7t2.a.a | $2$ | $ 2^{2} \cdot 29^{2}$ | 7.1.38068692544.1 | $D_{7}$ (as 7T2) | $1$ | $0$ | |
| * | 2.3364.14t8.b.d | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| * | 2.3364.14t8.b.f | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| * | 2.3364.14t8.b.a | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.3364.14t8.a.d | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (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 *.