Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 - 345*x^8 + 720*x^7 + 33292*x^6 - 77348*x^5 - 795480*x^4 + 1563624*x^3 + 4991153*x^2 - 5404190*x - 10994849)
gp: K = bnfinit(y^10 - 2*y^9 - 345*y^8 + 720*y^7 + 33292*y^6 - 77348*y^5 - 795480*y^4 + 1563624*y^3 + 4991153*y^2 - 5404190*y - 10994849, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 2*x^9 - 345*x^8 + 720*x^7 + 33292*x^6 - 77348*x^5 - 795480*x^4 + 1563624*x^3 + 4991153*x^2 - 5404190*x - 10994849);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 2*x^9 - 345*x^8 + 720*x^7 + 33292*x^6 - 77348*x^5 - 795480*x^4 + 1563624*x^3 + 4991153*x^2 - 5404190*x - 10994849)
\( x^{10} - 2 x^{9} - 345 x^{8} + 720 x^{7} + 33292 x^{6} - 77348 x^{5} - 795480 x^{4} + 1563624 x^{3} + \cdots - 10994849 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[10, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(23300317255158046392320000\)
\(\medspace = 2^{21}\cdot 5^{4}\cdot 71^{4}\cdot 26449^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(344.14\) | 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: | $2^{9/4}5^{1/2}71^{4/5}26449^{1/2}\approx 52361.707423576365$ | ||
| Ramified primes: |
\(2\), \(5\), \(71\), \(26449\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{14}a^{8}+\frac{1}{7}a^{7}+\frac{1}{7}a^{6}+\frac{3}{7}a^{5}+\frac{1}{7}a^{4}+\frac{3}{7}a^{2}+\frac{1}{7}a-\frac{3}{14}$, $\frac{1}{24\!\cdots\!28}a^{9}+\frac{70\!\cdots\!33}{34\!\cdots\!04}a^{8}+\frac{20\!\cdots\!27}{12\!\cdots\!64}a^{7}-\frac{12\!\cdots\!65}{12\!\cdots\!64}a^{6}-\frac{11\!\cdots\!15}{12\!\cdots\!64}a^{5}-\frac{67\!\cdots\!41}{12\!\cdots\!64}a^{4}+\frac{12\!\cdots\!59}{12\!\cdots\!64}a^{3}-\frac{60\!\cdots\!21}{12\!\cdots\!64}a^{2}-\frac{67\!\cdots\!33}{49\!\cdots\!72}a+\frac{67\!\cdots\!29}{24\!\cdots\!28}$
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: | $9$ | 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: |
$\frac{529689496193}{43\!\cdots\!32}a^{9}-\frac{1119994967803}{61\!\cdots\!76}a^{8}-\frac{141387944528709}{21\!\cdots\!16}a^{7}+\frac{12\!\cdots\!55}{21\!\cdots\!16}a^{6}+\frac{25\!\cdots\!01}{21\!\cdots\!16}a^{5}-\frac{11\!\cdots\!41}{21\!\cdots\!16}a^{4}-\frac{15\!\cdots\!09}{21\!\cdots\!16}a^{3}+\frac{31\!\cdots\!07}{21\!\cdots\!16}a^{2}+\frac{57\!\cdots\!13}{61\!\cdots\!76}a+\frac{19\!\cdots\!85}{43\!\cdots\!32}$, $\frac{49\!\cdots\!79}{24\!\cdots\!28}a^{9}-\frac{26\!\cdots\!93}{34\!\cdots\!04}a^{8}-\frac{83\!\cdots\!83}{12\!\cdots\!64}a^{7}+\frac{31\!\cdots\!73}{12\!\cdots\!64}a^{6}+\frac{75\!\cdots\!47}{12\!\cdots\!64}a^{5}-\frac{30\!\cdots\!07}{12\!\cdots\!64}a^{4}-\frac{13\!\cdots\!19}{12\!\cdots\!64}a^{3}+\frac{58\!\cdots\!29}{12\!\cdots\!64}a^{2}+\frac{74\!\cdots\!01}{49\!\cdots\!72}a-\frac{29\!\cdots\!13}{24\!\cdots\!28}$, $\frac{37\!\cdots\!43}{30\!\cdots\!16}a^{9}+\frac{55\!\cdots\!47}{43\!\cdots\!88}a^{8}-\frac{64\!\cdots\!11}{15\!\cdots\!08}a^{7}-\frac{59\!\cdots\!31}{15\!\cdots\!08}a^{6}+\frac{61\!\cdots\!67}{15\!\cdots\!08}a^{5}+\frac{39\!\cdots\!25}{15\!\cdots\!08}a^{4}-\frac{13\!\cdots\!71}{15\!\cdots\!08}a^{3}-\frac{12\!\cdots\!55}{15\!\cdots\!08}a^{2}+\frac{15\!\cdots\!91}{43\!\cdots\!88}a+\frac{13\!\cdots\!67}{30\!\cdots\!16}$, $\frac{49\!\cdots\!09}{12\!\cdots\!64}a^{9}-\frac{35\!\cdots\!15}{17\!\cdots\!52}a^{8}-\frac{80\!\cdots\!53}{61\!\cdots\!32}a^{7}+\frac{41\!\cdots\!27}{61\!\cdots\!32}a^{6}+\frac{65\!\cdots\!49}{61\!\cdots\!32}a^{5}-\frac{36\!\cdots\!09}{61\!\cdots\!32}a^{4}-\frac{49\!\cdots\!89}{61\!\cdots\!32}a^{3}+\frac{38\!\cdots\!03}{61\!\cdots\!32}a^{2}+\frac{11\!\cdots\!25}{17\!\cdots\!52}a-\frac{19\!\cdots\!27}{12\!\cdots\!64}$, $\frac{16\!\cdots\!15}{30\!\cdots\!16}a^{9}-\frac{89\!\cdots\!85}{43\!\cdots\!88}a^{8}-\frac{28\!\cdots\!51}{15\!\cdots\!08}a^{7}+\frac{11\!\cdots\!41}{15\!\cdots\!08}a^{6}+\frac{26\!\cdots\!43}{15\!\cdots\!08}a^{5}-\frac{11\!\cdots\!19}{15\!\cdots\!08}a^{4}-\frac{48\!\cdots\!79}{15\!\cdots\!08}a^{3}+\frac{21\!\cdots\!77}{15\!\cdots\!08}a^{2}+\frac{11\!\cdots\!71}{43\!\cdots\!88}a-\frac{10\!\cdots\!17}{30\!\cdots\!16}$, $\frac{28\!\cdots\!05}{34\!\cdots\!04}a^{9}-\frac{11\!\cdots\!77}{34\!\cdots\!04}a^{8}-\frac{48\!\cdots\!49}{17\!\cdots\!52}a^{7}+\frac{28\!\cdots\!41}{24\!\cdots\!36}a^{6}+\frac{44\!\cdots\!77}{17\!\cdots\!52}a^{5}-\frac{19\!\cdots\!01}{17\!\cdots\!52}a^{4}-\frac{77\!\cdots\!45}{17\!\cdots\!52}a^{3}+\frac{37\!\cdots\!99}{17\!\cdots\!52}a^{2}+\frac{20\!\cdots\!15}{34\!\cdots\!04}a-\frac{16\!\cdots\!07}{34\!\cdots\!04}$, $\frac{22\!\cdots\!13}{12\!\cdots\!64}a^{9}+\frac{30\!\cdots\!65}{17\!\cdots\!52}a^{8}-\frac{28\!\cdots\!93}{61\!\cdots\!32}a^{7}-\frac{27\!\cdots\!69}{61\!\cdots\!32}a^{6}+\frac{12\!\cdots\!05}{61\!\cdots\!32}a^{5}+\frac{11\!\cdots\!83}{61\!\cdots\!32}a^{4}+\frac{26\!\cdots\!31}{61\!\cdots\!32}a^{3}-\frac{79\!\cdots\!69}{61\!\cdots\!32}a^{2}-\frac{33\!\cdots\!95}{17\!\cdots\!52}a-\frac{70\!\cdots\!91}{12\!\cdots\!64}$, $\frac{26\!\cdots\!77}{61\!\cdots\!32}a^{9}-\frac{51\!\cdots\!91}{87\!\cdots\!76}a^{8}-\frac{25\!\cdots\!53}{30\!\cdots\!16}a^{7}+\frac{38\!\cdots\!03}{30\!\cdots\!16}a^{6}+\frac{20\!\cdots\!33}{30\!\cdots\!16}a^{5}-\frac{10\!\cdots\!37}{30\!\cdots\!16}a^{4}+\frac{12\!\cdots\!43}{30\!\cdots\!16}a^{3}+\frac{63\!\cdots\!27}{30\!\cdots\!16}a^{2}-\frac{14\!\cdots\!63}{87\!\cdots\!76}a-\frac{26\!\cdots\!47}{61\!\cdots\!32}$, $\frac{19\!\cdots\!23}{61\!\cdots\!32}a^{9}+\frac{43\!\cdots\!07}{87\!\cdots\!76}a^{8}-\frac{33\!\cdots\!31}{30\!\cdots\!16}a^{7}-\frac{52\!\cdots\!31}{30\!\cdots\!16}a^{6}+\frac{31\!\cdots\!55}{30\!\cdots\!16}a^{5}+\frac{42\!\cdots\!37}{30\!\cdots\!16}a^{4}-\frac{69\!\cdots\!71}{30\!\cdots\!16}a^{3}-\frac{10\!\cdots\!75}{30\!\cdots\!16}a^{2}+\frac{64\!\cdots\!95}{87\!\cdots\!76}a+\frac{70\!\cdots\!59}{61\!\cdots\!32}$
| 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: | \( 4673862280.11 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 4673862280.11 \cdot 3}{2\cdot\sqrt{23300317255158046392320000}}\cr\approx \mathstrut & 1.48725764438 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 - 345*x^8 + 720*x^7 + 33292*x^6 - 77348*x^5 - 795480*x^4 + 1563624*x^3 + 4991153*x^2 - 5404190*x - 10994849)
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^10 - 2*x^9 - 345*x^8 + 720*x^7 + 33292*x^6 - 77348*x^5 - 795480*x^4 + 1563624*x^3 + 4991153*x^2 - 5404190*x - 10994849, 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^10 - 2*x^9 - 345*x^8 + 720*x^7 + 33292*x^6 - 77348*x^5 - 795480*x^4 + 1563624*x^3 + 4991153*x^2 - 5404190*x - 10994849);
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^10 - 2*x^9 - 345*x^8 + 720*x^7 + 33292*x^6 - 77348*x^5 - 795480*x^4 + 1563624*x^3 + 4991153*x^2 - 5404190*x - 10994849);
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
$\SOPlus(4,4)$ (as 10T40):
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 non-solvable group of order 7200 |
| The 20 conjugacy class representatives for $A_5 \wr C_2$ |
| Character table for $A_5 \wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \) |
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 12 sibling: | data not computed |
| Degree 20 sibling: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 25 sibling: | data not computed |
| Degree 30 sibling: | data not computed |
| Degree 36 sibling: | data not computed |
| Degree 40 sibling: | 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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$ | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.8.18.60 | $x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{5} + 2 x^{4} + 4 x^{3} + 4 x^{2} + 2$ | $8$ | $1$ | $18$ | $A_4\times C_2$ | $[2, 2, 3]^{3}$ | |
|
\(5\)
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(71\)
| $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 71.3.0.1 | $x^{3} + 4 x + 64$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 71.5.4.4 | $x^{5} + 355$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
|
\(26449\)
| $\Q_{26449}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{26449}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{26449}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
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.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 6.722...000.24t9631.a.a | $6$ | $ 2^{15} \cdot 5^{4} \cdot 71^{2} \cdot 26449^{2}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $6$ | |
| 6.722...000.24t9631.a.b | $6$ | $ 2^{15} \cdot 5^{4} \cdot 71^{2} \cdot 26449^{2}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $6$ | |
| * | 8.291...000.10t40.a.a | $8$ | $ 2^{18} \cdot 5^{4} \cdot 71^{4} \cdot 26449^{2}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $8$ |
| 9.459...000.72.a.a | $9$ | $ 2^{24} \cdot 5^{4} \cdot 71^{6} \cdot 26449^{6}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $9$ | |
| 9.459...000.72.a.b | $9$ | $ 2^{24} \cdot 5^{4} \cdot 71^{6} \cdot 26449^{6}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $9$ | |
| 9.898...000.60.a.a | $9$ | $ 2^{15} \cdot 5^{4} \cdot 71^{6} \cdot 26449^{6}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $9$ | |
| 9.898...000.60.a.b | $9$ | $ 2^{15} \cdot 5^{4} \cdot 71^{6} \cdot 26449^{6}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $9$ | |
| 10.233...000.12t269.a.a | $10$ | $ 2^{21} \cdot 5^{4} \cdot 71^{4} \cdot 26449^{2}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $10$ | |
| 16.418...000.25t88.a.a | $16$ | $ 2^{30} \cdot 5^{8} \cdot 71^{16} \cdot 26449^{8}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $16$ | |
| 16.171...000.50.a.a | $16$ | $ 2^{42} \cdot 5^{8} \cdot 71^{16} \cdot 26449^{8}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $16$ | |
| 18.412...000.60.a.a | $18$ | $ 2^{39} \cdot 5^{8} \cdot 71^{12} \cdot 26449^{12}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $18$ | |
| 24.382...000.120.a.a | $24$ | $ 2^{54} \cdot 5^{12} \cdot 71^{20} \cdot 26449^{14}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $24$ | |
| 24.382...000.120.a.b | $24$ | $ 2^{54} \cdot 5^{12} \cdot 71^{20} \cdot 26449^{14}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $24$ | |
| 25.121...000.36t7075.a.a | $25$ | $ 2^{48} \cdot 5^{12} \cdot 71^{20} \cdot 26449^{10}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $25$ | |
| 25.399...000.72.a.a | $25$ | $ 2^{63} \cdot 5^{12} \cdot 71^{20} \cdot 26449^{10}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $25$ | |
| 30.275...000.144.a.a | $30$ | $ 2^{69} \cdot 5^{16} \cdot 71^{22} \cdot 26449^{16}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $30$ | |
| 30.275...000.144.a.b | $30$ | $ 2^{69} \cdot 5^{16} \cdot 71^{22} \cdot 26449^{16}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $30$ | |
| 40.209...000.60.a.a | $40$ | $ 2^{90} \cdot 5^{20} \cdot 71^{36} \cdot 26449^{18}$ | 10.10.23300317255158046392320000.1 | $A_5 \wr C_2$ (as 10T40) | $1$ | $40$ |
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 *.