Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 + 22*x^13 - 38*x^12 + 86*x^11 - 206*x^10 + 364*x^9 - 800*x^8 - 1462*x^7 + 615*x^6 - 6007*x^5 + 2169*x^4 - 965*x^3 - 13194*x^2 + 6050*x - 19375)
gp: K = bnfinit(y^15 - 7*y^14 + 22*y^13 - 38*y^12 + 86*y^11 - 206*y^10 + 364*y^9 - 800*y^8 - 1462*y^7 + 615*y^6 - 6007*y^5 + 2169*y^4 - 965*y^3 - 13194*y^2 + 6050*y - 19375, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 7*x^14 + 22*x^13 - 38*x^12 + 86*x^11 - 206*x^10 + 364*x^9 - 800*x^8 - 1462*x^7 + 615*x^6 - 6007*x^5 + 2169*x^4 - 965*x^3 - 13194*x^2 + 6050*x - 19375);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 7*x^14 + 22*x^13 - 38*x^12 + 86*x^11 - 206*x^10 + 364*x^9 - 800*x^8 - 1462*x^7 + 615*x^6 - 6007*x^5 + 2169*x^4 - 965*x^3 - 13194*x^2 + 6050*x - 19375)
\( x^{15} - 7 x^{14} + 22 x^{13} - 38 x^{12} + 86 x^{11} - 206 x^{10} + 364 x^{9} - 800 x^{8} + \cdots - 19375 \)
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: |
\(-345408215017012205401343\)
\(\medspace = -\,7^{10}\cdot 11^{7}\cdot 13^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(37.09\) | 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: | $7^{2/3}11^{1/2}13^{1/2}\approx 43.758931819174485$ | ||
| Ramified primes: |
\(7\), \(11\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-143}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{5}a$, $\frac{1}{5}a^{10}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a$, $\frac{1}{55}a^{11}+\frac{4}{55}a^{10}-\frac{4}{55}a^{9}-\frac{21}{55}a^{8}+\frac{2}{11}a^{7}-\frac{6}{55}a^{6}+\frac{2}{11}a^{5}+\frac{1}{55}a^{4}-\frac{2}{55}a^{3}+\frac{19}{55}a^{2}+\frac{2}{5}a+\frac{2}{11}$, $\frac{1}{55}a^{12}+\frac{2}{55}a^{10}-\frac{1}{11}a^{9}+\frac{17}{55}a^{8}-\frac{24}{55}a^{7}+\frac{1}{55}a^{6}+\frac{27}{55}a^{5}+\frac{27}{55}a^{4}+\frac{27}{55}a^{3}+\frac{1}{55}a^{2}-\frac{1}{55}a+\frac{3}{11}$, $\frac{1}{55}a^{13}-\frac{2}{55}a^{10}+\frac{3}{55}a^{9}-\frac{3}{11}a^{8}-\frac{8}{55}a^{7}-\frac{27}{55}a^{6}-\frac{4}{55}a^{5}+\frac{14}{55}a^{4}+\frac{27}{55}a^{3}-\frac{17}{55}a^{2}+\frac{4}{55}a-\frac{4}{11}$, $\frac{1}{63\!\cdots\!25}a^{14}+\frac{22\!\cdots\!98}{63\!\cdots\!25}a^{13}+\frac{49\!\cdots\!57}{63\!\cdots\!25}a^{12}+\frac{49\!\cdots\!27}{63\!\cdots\!25}a^{11}+\frac{26\!\cdots\!31}{63\!\cdots\!25}a^{10}-\frac{15\!\cdots\!96}{63\!\cdots\!25}a^{9}-\frac{12\!\cdots\!76}{63\!\cdots\!25}a^{8}-\frac{31\!\cdots\!76}{12\!\cdots\!05}a^{7}-\frac{83\!\cdots\!22}{63\!\cdots\!25}a^{6}-\frac{59\!\cdots\!43}{19\!\cdots\!91}a^{5}+\frac{14\!\cdots\!73}{63\!\cdots\!25}a^{4}-\frac{27\!\cdots\!61}{63\!\cdots\!25}a^{3}+\frac{46\!\cdots\!28}{12\!\cdots\!05}a^{2}+\frac{49\!\cdots\!16}{63\!\cdots\!25}a-\frac{66\!\cdots\!85}{25\!\cdots\!21}$
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: | $7$ | 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: |
$\frac{22\!\cdots\!31}{48\!\cdots\!75}a^{14}-\frac{98\!\cdots\!52}{48\!\cdots\!75}a^{13}+\frac{88\!\cdots\!42}{48\!\cdots\!75}a^{12}+\frac{35\!\cdots\!32}{48\!\cdots\!75}a^{11}+\frac{22\!\cdots\!26}{48\!\cdots\!75}a^{10}-\frac{12\!\cdots\!96}{48\!\cdots\!75}a^{9}-\frac{20\!\cdots\!76}{48\!\cdots\!75}a^{8}+\frac{34\!\cdots\!59}{96\!\cdots\!55}a^{7}-\frac{69\!\cdots\!37}{48\!\cdots\!75}a^{6}-\frac{17\!\cdots\!78}{96\!\cdots\!55}a^{5}-\frac{49\!\cdots\!32}{48\!\cdots\!75}a^{4}-\frac{13\!\cdots\!66}{48\!\cdots\!75}a^{3}-\frac{12\!\cdots\!51}{96\!\cdots\!55}a^{2}-\frac{89\!\cdots\!99}{48\!\cdots\!75}a-\frac{20\!\cdots\!66}{19\!\cdots\!91}$, $\frac{36\!\cdots\!37}{48\!\cdots\!75}a^{14}-\frac{18\!\cdots\!64}{48\!\cdots\!75}a^{13}+\frac{23\!\cdots\!44}{48\!\cdots\!75}a^{12}+\frac{42\!\cdots\!89}{48\!\cdots\!75}a^{11}+\frac{44\!\cdots\!17}{48\!\cdots\!75}a^{10}-\frac{32\!\cdots\!77}{48\!\cdots\!75}a^{9}+\frac{81\!\cdots\!13}{48\!\cdots\!75}a^{8}-\frac{97\!\cdots\!89}{96\!\cdots\!55}a^{7}-\frac{10\!\cdots\!49}{48\!\cdots\!75}a^{6}-\frac{23\!\cdots\!51}{96\!\cdots\!55}a^{5}+\frac{19\!\cdots\!91}{48\!\cdots\!75}a^{4}-\frac{17\!\cdots\!77}{48\!\cdots\!75}a^{3}+\frac{18\!\cdots\!76}{19\!\cdots\!91}a^{2}+\frac{24\!\cdots\!32}{48\!\cdots\!75}a-\frac{16\!\cdots\!98}{19\!\cdots\!91}$, $\frac{6246985794080}{25\!\cdots\!51}a^{14}-\frac{366663832764178}{12\!\cdots\!55}a^{13}+\frac{16\!\cdots\!11}{12\!\cdots\!55}a^{12}-\frac{39\!\cdots\!96}{12\!\cdots\!55}a^{11}+\frac{82\!\cdots\!84}{12\!\cdots\!55}a^{10}-\frac{46\!\cdots\!47}{25\!\cdots\!51}a^{9}+\frac{48\!\cdots\!93}{12\!\cdots\!55}a^{8}-\frac{18\!\cdots\!52}{25\!\cdots\!51}a^{7}+\frac{14\!\cdots\!83}{12\!\cdots\!55}a^{6}+\frac{764974907676399}{98\!\cdots\!05}a^{5}+\frac{30\!\cdots\!14}{12\!\cdots\!55}a^{4}+\frac{10\!\cdots\!48}{12\!\cdots\!55}a^{3}+\frac{51\!\cdots\!73}{12\!\cdots\!55}a^{2}+\frac{20\!\cdots\!30}{25\!\cdots\!51}a+\frac{98\!\cdots\!82}{25\!\cdots\!51}$, $\frac{27\!\cdots\!62}{63\!\cdots\!25}a^{14}-\frac{24\!\cdots\!29}{63\!\cdots\!25}a^{13}-\frac{11\!\cdots\!26}{63\!\cdots\!25}a^{12}+\frac{72\!\cdots\!64}{63\!\cdots\!25}a^{11}-\frac{19\!\cdots\!33}{63\!\cdots\!25}a^{10}+\frac{41\!\cdots\!83}{63\!\cdots\!25}a^{9}-\frac{98\!\cdots\!07}{63\!\cdots\!25}a^{8}+\frac{46\!\cdots\!43}{12\!\cdots\!05}a^{7}-\frac{57\!\cdots\!24}{63\!\cdots\!25}a^{6}+\frac{88\!\cdots\!26}{96\!\cdots\!55}a^{5}+\frac{17\!\cdots\!81}{63\!\cdots\!25}a^{4}-\frac{11\!\cdots\!02}{63\!\cdots\!25}a^{3}+\frac{38\!\cdots\!86}{12\!\cdots\!05}a^{2}-\frac{22\!\cdots\!53}{63\!\cdots\!25}a+\frac{50\!\cdots\!12}{25\!\cdots\!21}$, $\frac{51\!\cdots\!69}{25\!\cdots\!21}a^{14}-\frac{13\!\cdots\!18}{12\!\cdots\!05}a^{13}+\frac{21\!\cdots\!02}{12\!\cdots\!05}a^{12}+\frac{24\!\cdots\!92}{12\!\cdots\!05}a^{11}+\frac{13\!\cdots\!04}{12\!\cdots\!05}a^{10}-\frac{51\!\cdots\!98}{12\!\cdots\!05}a^{9}+\frac{13\!\cdots\!73}{25\!\cdots\!21}a^{8}-\frac{15\!\cdots\!28}{12\!\cdots\!05}a^{7}-\frac{37\!\cdots\!79}{12\!\cdots\!05}a^{6}-\frac{99\!\cdots\!33}{96\!\cdots\!55}a^{5}+\frac{88\!\cdots\!43}{12\!\cdots\!05}a^{4}-\frac{21\!\cdots\!08}{25\!\cdots\!21}a^{3}-\frac{17\!\cdots\!28}{12\!\cdots\!05}a^{2}+\frac{14\!\cdots\!97}{12\!\cdots\!05}a-\frac{13\!\cdots\!07}{25\!\cdots\!21}$, $\frac{91\!\cdots\!26}{57\!\cdots\!75}a^{14}-\frac{83\!\cdots\!77}{57\!\cdots\!75}a^{13}+\frac{31\!\cdots\!12}{57\!\cdots\!75}a^{12}-\frac{55\!\cdots\!68}{57\!\cdots\!75}a^{11}+\frac{58\!\cdots\!41}{57\!\cdots\!75}a^{10}-\frac{14\!\cdots\!31}{57\!\cdots\!75}a^{9}+\frac{46\!\cdots\!09}{57\!\cdots\!75}a^{8}-\frac{22\!\cdots\!04}{11\!\cdots\!55}a^{7}-\frac{89\!\cdots\!32}{57\!\cdots\!75}a^{6}+\frac{60\!\cdots\!14}{88\!\cdots\!05}a^{5}-\frac{37\!\cdots\!97}{52\!\cdots\!25}a^{4}+\frac{21\!\cdots\!89}{52\!\cdots\!25}a^{3}+\frac{70\!\cdots\!36}{11\!\cdots\!55}a^{2}-\frac{10\!\cdots\!84}{57\!\cdots\!75}a+\frac{55\!\cdots\!22}{23\!\cdots\!11}$, $\frac{17\!\cdots\!17}{63\!\cdots\!25}a^{14}-\frac{87\!\cdots\!04}{63\!\cdots\!25}a^{13}+\frac{10\!\cdots\!69}{63\!\cdots\!25}a^{12}+\frac{16\!\cdots\!89}{63\!\cdots\!25}a^{11}+\frac{49\!\cdots\!12}{63\!\cdots\!25}a^{10}-\frac{16\!\cdots\!37}{63\!\cdots\!25}a^{9}-\frac{21\!\cdots\!22}{63\!\cdots\!25}a^{8}-\frac{14\!\cdots\!24}{25\!\cdots\!21}a^{7}-\frac{34\!\cdots\!14}{63\!\cdots\!25}a^{6}-\frac{20\!\cdots\!14}{96\!\cdots\!55}a^{5}+\frac{11\!\cdots\!91}{63\!\cdots\!25}a^{4}-\frac{62\!\cdots\!37}{63\!\cdots\!25}a^{3}-\frac{33\!\cdots\!37}{12\!\cdots\!05}a^{2}-\frac{62\!\cdots\!23}{63\!\cdots\!25}a-\frac{57\!\cdots\!63}{25\!\cdots\!21}$
| 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: | \( 478768.310095 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 478768.310095 \cdot 3}{2\cdot\sqrt{345408215017012205401343}}\cr\approx \mathstrut & 0.944798942600 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 + 22*x^13 - 38*x^12 + 86*x^11 - 206*x^10 + 364*x^9 - 800*x^8 - 1462*x^7 + 615*x^6 - 6007*x^5 + 2169*x^4 - 965*x^3 - 13194*x^2 + 6050*x - 19375)
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 - 7*x^14 + 22*x^13 - 38*x^12 + 86*x^11 - 206*x^10 + 364*x^9 - 800*x^8 - 1462*x^7 + 615*x^6 - 6007*x^5 + 2169*x^4 - 965*x^3 - 13194*x^2 + 6050*x - 19375, 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 - 7*x^14 + 22*x^13 - 38*x^12 + 86*x^11 - 206*x^10 + 364*x^9 - 800*x^8 - 1462*x^7 + 615*x^6 - 6007*x^5 + 2169*x^4 - 965*x^3 - 13194*x^2 + 6050*x - 19375);
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 - 7*x^14 + 22*x^13 - 38*x^12 + 86*x^11 - 206*x^10 + 364*x^9 - 800*x^8 - 1462*x^7 + 615*x^6 - 6007*x^5 + 2169*x^4 - 965*x^3 - 13194*x^2 + 6050*x - 19375);
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
| 3.1.7007.1, 5.1.20449.1 |
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 | $15$ | $15$ | ${\href{/padicField/5.2.0.1}{2} }^{7}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | R | R | ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }^{3}$ | ${\href{/padicField/23.5.0.1}{5} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15$ | ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.5.0.1}{5} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.15.10.2 | $x^{15} + 2401 x^{3} - 67228$ | $3$ | $5$ | $10$ | $C_{15}$ | $[\ ]_{3}^{5}$ |
|
\(11\)
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.143.2t1.a.a | $1$ | $ 11 \cdot 13 $ | \(\Q(\sqrt{-143}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 2.7007.3t2.a.a | $2$ | $ 7^{2} \cdot 11 \cdot 13 $ | 3.1.7007.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.143.5t2.a.b | $2$ | $ 11 \cdot 13 $ | 5.1.20449.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.143.5t2.a.a | $2$ | $ 11 \cdot 13 $ | 5.1.20449.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.7007.15t2.a.b | $2$ | $ 7^{2} \cdot 11 \cdot 13 $ | 15.1.345408215017012205401343.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.7007.15t2.a.d | $2$ | $ 7^{2} \cdot 11 \cdot 13 $ | 15.1.345408215017012205401343.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.7007.15t2.a.a | $2$ | $ 7^{2} \cdot 11 \cdot 13 $ | 15.1.345408215017012205401343.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.7007.15t2.a.c | $2$ | $ 7^{2} \cdot 11 \cdot 13 $ | 15.1.345408215017012205401343.1 | $D_{15}$ (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 *.