Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 29*x^13 - 119*x^12 + 396*x^11 - 1132*x^10 + 2906*x^9 - 6522*x^8 + 13105*x^7 - 24007*x^6 + 38721*x^5 - 52199*x^4 + 52599*x^3 - 37137*x^2 + 14859*x - 2421)
gp: K = bnfinit(y^15 - 6*y^14 + 29*y^13 - 119*y^12 + 396*y^11 - 1132*y^10 + 2906*y^9 - 6522*y^8 + 13105*y^7 - 24007*y^6 + 38721*y^5 - 52199*y^4 + 52599*y^3 - 37137*y^2 + 14859*y - 2421, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 6*x^14 + 29*x^13 - 119*x^12 + 396*x^11 - 1132*x^10 + 2906*x^9 - 6522*x^8 + 13105*x^7 - 24007*x^6 + 38721*x^5 - 52199*x^4 + 52599*x^3 - 37137*x^2 + 14859*x - 2421);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 6*x^14 + 29*x^13 - 119*x^12 + 396*x^11 - 1132*x^10 + 2906*x^9 - 6522*x^8 + 13105*x^7 - 24007*x^6 + 38721*x^5 - 52199*x^4 + 52599*x^3 - 37137*x^2 + 14859*x - 2421)
\( x^{15} - 6 x^{14} + 29 x^{13} - 119 x^{12} + 396 x^{11} - 1132 x^{10} + 2906 x^{9} - 6522 x^{8} + \cdots - 2421 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-8450344007000266933623879\)
\(\medspace = -\,3^{7}\cdot 1213^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(45.90\) | 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: | $3^{1/2}1213^{1/2}\approx 60.32412452742269$ | ||
| Ramified primes: |
\(3\), \(1213\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3639}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{9}-\frac{1}{3}a^{5}-\frac{4}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{10}-\frac{1}{9}a^{4}$, $\frac{1}{9}a^{11}-\frac{1}{9}a^{5}$, $\frac{1}{9}a^{12}-\frac{1}{9}a^{6}$, $\frac{1}{81}a^{13}+\frac{4}{81}a^{12}-\frac{4}{81}a^{11}-\frac{1}{81}a^{10}+\frac{1}{27}a^{9}-\frac{1}{27}a^{8}+\frac{2}{81}a^{7}+\frac{8}{81}a^{6}+\frac{25}{81}a^{5}-\frac{32}{81}a^{4}-\frac{4}{9}a^{3}-\frac{1}{27}a^{2}-\frac{1}{9}a+\frac{1}{9}$, $\frac{1}{30\!\cdots\!31}a^{14}+\frac{552779285442629}{30\!\cdots\!31}a^{13}+\frac{549848504754818}{33\!\cdots\!59}a^{12}-\frac{27017080172585}{942869095121349}a^{11}-\frac{884867963771006}{27\!\cdots\!21}a^{10}+\frac{398355002683082}{33\!\cdots\!59}a^{9}-\frac{16\!\cdots\!07}{27\!\cdots\!21}a^{8}+\frac{12\!\cdots\!34}{30\!\cdots\!31}a^{7}+\frac{42\!\cdots\!04}{10\!\cdots\!77}a^{6}-\frac{16\!\cdots\!96}{30\!\cdots\!31}a^{5}-\frac{66\!\cdots\!16}{30\!\cdots\!31}a^{4}+\frac{10\!\cdots\!91}{10\!\cdots\!77}a^{3}+\frac{49\!\cdots\!60}{10\!\cdots\!77}a^{2}-\frac{27\!\cdots\!34}{11\!\cdots\!53}a-\frac{93\!\cdots\!76}{33\!\cdots\!59}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
$C_{7}$, which has order $7$ (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: | $7$ | 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: |
$\frac{110098422693676}{30\!\cdots\!31}a^{14}-\frac{871031322155096}{30\!\cdots\!31}a^{13}+\frac{36\!\cdots\!48}{30\!\cdots\!31}a^{12}-\frac{48421391659897}{942869095121349}a^{11}+\frac{15\!\cdots\!11}{91\!\cdots\!07}a^{10}-\frac{46\!\cdots\!79}{10\!\cdots\!77}a^{9}+\frac{31\!\cdots\!98}{27\!\cdots\!21}a^{8}-\frac{78\!\cdots\!98}{30\!\cdots\!31}a^{7}+\frac{14\!\cdots\!75}{30\!\cdots\!31}a^{6}-\frac{26\!\cdots\!14}{30\!\cdots\!31}a^{5}+\frac{14\!\cdots\!53}{10\!\cdots\!77}a^{4}-\frac{16\!\cdots\!67}{10\!\cdots\!77}a^{3}+\frac{14\!\cdots\!51}{11\!\cdots\!53}a^{2}-\frac{15\!\cdots\!12}{33\!\cdots\!59}a+\frac{67\!\cdots\!20}{11\!\cdots\!53}$, $\frac{56\!\cdots\!55}{30\!\cdots\!31}a^{14}-\frac{10\!\cdots\!44}{10\!\cdots\!77}a^{13}+\frac{14\!\cdots\!85}{30\!\cdots\!31}a^{12}-\frac{619440588067219}{314289698373783}a^{11}+\frac{17\!\cdots\!62}{27\!\cdots\!21}a^{10}-\frac{17\!\cdots\!24}{10\!\cdots\!77}a^{9}+\frac{12\!\cdots\!16}{27\!\cdots\!21}a^{8}-\frac{98\!\cdots\!80}{10\!\cdots\!77}a^{7}+\frac{58\!\cdots\!38}{30\!\cdots\!31}a^{6}-\frac{34\!\cdots\!44}{10\!\cdots\!77}a^{5}+\frac{16\!\cdots\!90}{30\!\cdots\!31}a^{4}-\frac{68\!\cdots\!75}{10\!\cdots\!77}a^{3}+\frac{61\!\cdots\!21}{10\!\cdots\!77}a^{2}-\frac{12\!\cdots\!26}{33\!\cdots\!59}a+\frac{26\!\cdots\!58}{33\!\cdots\!59}$, $\frac{643609340621560}{30\!\cdots\!31}a^{14}-\frac{10\!\cdots\!63}{10\!\cdots\!77}a^{13}+\frac{14\!\cdots\!09}{30\!\cdots\!31}a^{12}-\frac{61075729864892}{314289698373783}a^{11}+\frac{16\!\cdots\!70}{27\!\cdots\!21}a^{10}-\frac{16\!\cdots\!17}{10\!\cdots\!77}a^{9}+\frac{11\!\cdots\!15}{27\!\cdots\!21}a^{8}-\frac{86\!\cdots\!03}{10\!\cdots\!77}a^{7}+\frac{49\!\cdots\!20}{30\!\cdots\!31}a^{6}-\frac{29\!\cdots\!15}{10\!\cdots\!77}a^{5}+\frac{13\!\cdots\!46}{30\!\cdots\!31}a^{4}-\frac{51\!\cdots\!47}{10\!\cdots\!77}a^{3}+\frac{37\!\cdots\!88}{10\!\cdots\!77}a^{2}-\frac{51\!\cdots\!27}{33\!\cdots\!59}a-\frac{12\!\cdots\!58}{33\!\cdots\!59}$, $\frac{10\!\cdots\!82}{10\!\cdots\!77}a^{14}-\frac{18\!\cdots\!85}{33\!\cdots\!59}a^{13}+\frac{26\!\cdots\!84}{10\!\cdots\!77}a^{12}-\frac{11\!\cdots\!24}{104763232791261}a^{11}+\frac{31\!\cdots\!97}{91\!\cdots\!07}a^{10}-\frac{32\!\cdots\!92}{33\!\cdots\!59}a^{9}+\frac{22\!\cdots\!16}{91\!\cdots\!07}a^{8}-\frac{17\!\cdots\!10}{33\!\cdots\!59}a^{7}+\frac{10\!\cdots\!62}{10\!\cdots\!77}a^{6}-\frac{63\!\cdots\!02}{33\!\cdots\!59}a^{5}+\frac{29\!\cdots\!41}{10\!\cdots\!77}a^{4}-\frac{12\!\cdots\!31}{33\!\cdots\!59}a^{3}+\frac{11\!\cdots\!81}{33\!\cdots\!59}a^{2}-\frac{22\!\cdots\!06}{11\!\cdots\!53}a+\frac{52\!\cdots\!43}{11\!\cdots\!53}$, $\frac{133902148977308}{30\!\cdots\!31}a^{14}-\frac{214882211894551}{10\!\cdots\!77}a^{13}+\frac{30\!\cdots\!75}{30\!\cdots\!31}a^{12}-\frac{12789276307789}{314289698373783}a^{11}+\frac{34\!\cdots\!90}{27\!\cdots\!21}a^{10}-\frac{34\!\cdots\!23}{10\!\cdots\!77}a^{9}+\frac{23\!\cdots\!28}{27\!\cdots\!21}a^{8}-\frac{17\!\cdots\!36}{10\!\cdots\!77}a^{7}+\frac{10\!\cdots\!99}{30\!\cdots\!31}a^{6}-\frac{60\!\cdots\!95}{10\!\cdots\!77}a^{5}+\frac{26\!\cdots\!18}{30\!\cdots\!31}a^{4}-\frac{10\!\cdots\!08}{10\!\cdots\!77}a^{3}+\frac{82\!\cdots\!90}{10\!\cdots\!77}a^{2}-\frac{16\!\cdots\!76}{33\!\cdots\!59}a+\frac{45\!\cdots\!31}{33\!\cdots\!59}$, $\frac{18\!\cdots\!38}{30\!\cdots\!31}a^{14}-\frac{501888842885077}{33\!\cdots\!59}a^{13}+\frac{20\!\cdots\!33}{30\!\cdots\!31}a^{12}-\frac{72102218808298}{314289698373783}a^{11}+\frac{11\!\cdots\!49}{27\!\cdots\!21}a^{10}-\frac{77\!\cdots\!22}{10\!\cdots\!77}a^{9}+\frac{33\!\cdots\!13}{27\!\cdots\!21}a^{8}+\frac{30\!\cdots\!52}{33\!\cdots\!59}a^{7}-\frac{16\!\cdots\!70}{30\!\cdots\!31}a^{6}+\frac{14\!\cdots\!67}{10\!\cdots\!77}a^{5}-\frac{13\!\cdots\!08}{30\!\cdots\!31}a^{4}+\frac{95\!\cdots\!30}{10\!\cdots\!77}a^{3}-\frac{14\!\cdots\!12}{10\!\cdots\!77}a^{2}+\frac{31\!\cdots\!19}{33\!\cdots\!59}a-\frac{66\!\cdots\!62}{33\!\cdots\!59}$, $\frac{16\!\cdots\!97}{30\!\cdots\!31}a^{14}-\frac{80\!\cdots\!66}{30\!\cdots\!31}a^{13}+\frac{42\!\cdots\!39}{33\!\cdots\!59}a^{12}-\frac{470658427720621}{942869095121349}a^{11}+\frac{42\!\cdots\!03}{27\!\cdots\!21}a^{10}-\frac{14\!\cdots\!93}{33\!\cdots\!59}a^{9}+\frac{28\!\cdots\!01}{27\!\cdots\!21}a^{8}-\frac{67\!\cdots\!92}{30\!\cdots\!31}a^{7}+\frac{43\!\cdots\!20}{10\!\cdots\!77}a^{6}-\frac{22\!\cdots\!04}{30\!\cdots\!31}a^{5}+\frac{33\!\cdots\!54}{30\!\cdots\!31}a^{4}-\frac{13\!\cdots\!07}{10\!\cdots\!77}a^{3}+\frac{97\!\cdots\!95}{10\!\cdots\!77}a^{2}-\frac{44\!\cdots\!12}{11\!\cdots\!53}a-\frac{29\!\cdots\!35}{33\!\cdots\!59}$
| 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: | \( 4974627.527237572 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 4974627.527237572 \cdot 7}{2\cdot\sqrt{8450344007000266933623879}}\cr\approx \mathstrut & 4.63106090571296 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 29*x^13 - 119*x^12 + 396*x^11 - 1132*x^10 + 2906*x^9 - 6522*x^8 + 13105*x^7 - 24007*x^6 + 38721*x^5 - 52199*x^4 + 52599*x^3 - 37137*x^2 + 14859*x - 2421)
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^15 - 6*x^14 + 29*x^13 - 119*x^12 + 396*x^11 - 1132*x^10 + 2906*x^9 - 6522*x^8 + 13105*x^7 - 24007*x^6 + 38721*x^5 - 52199*x^4 + 52599*x^3 - 37137*x^2 + 14859*x - 2421, 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^15 - 6*x^14 + 29*x^13 - 119*x^12 + 396*x^11 - 1132*x^10 + 2906*x^9 - 6522*x^8 + 13105*x^7 - 24007*x^6 + 38721*x^5 - 52199*x^4 + 52599*x^3 - 37137*x^2 + 14859*x - 2421);
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^15 - 6*x^14 + 29*x^13 - 119*x^12 + 396*x^11 - 1132*x^10 + 2906*x^9 - 6522*x^8 + 13105*x^7 - 24007*x^6 + 38721*x^5 - 52199*x^4 + 52599*x^3 - 37137*x^2 + 14859*x - 2421);
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 solvable group of order 30 |
| The 9 conjugacy class representatives for $D_{15}$ |
| Character table for $D_{15}$ |
Intermediate fields
| 3.1.3639.1, 5.1.13242321.1 |
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
| Galois closure: | deg 30 |
| 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 | ${\href{/padicField/2.5.0.1}{5} }^{3}$ | R | $15$ | $15$ | ${\href{/padicField/11.2.0.1}{2} }^{7}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $15$ | $15$ | ${\href{/padicField/19.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $15$ | ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.3.0.1}{3} }^{5}$ | ${\href{/padicField/43.3.0.1}{3} }^{5}$ | $15$ | ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.5.0.1}{5} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(1213\)
| $\Q_{1213}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.3639.2t1.a.a | $1$ | $ 3 \cdot 1213 $ | \(\Q(\sqrt{-3639}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 2.3639.3t2.a.a | $2$ | $ 3 \cdot 1213 $ | 3.1.3639.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.3639.5t2.a.b | $2$ | $ 3 \cdot 1213 $ | 5.1.13242321.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.3639.5t2.a.a | $2$ | $ 3 \cdot 1213 $ | 5.1.13242321.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.3639.15t2.a.a | $2$ | $ 3 \cdot 1213 $ | 15.1.8450344007000266933623879.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.3639.15t2.a.b | $2$ | $ 3 \cdot 1213 $ | 15.1.8450344007000266933623879.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.3639.15t2.a.c | $2$ | $ 3 \cdot 1213 $ | 15.1.8450344007000266933623879.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.3639.15t2.a.d | $2$ | $ 3 \cdot 1213 $ | 15.1.8450344007000266933623879.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
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 *.