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Properties

Label 30.2.493...181.1
Degree $30$
Signature $[2, 14]$
Discriminant $4.939\times 10^{51}$
Root discriminant \(52.86\)
Ramified primes $3,7,47$
Class number not computed
Class group not computed
Galois group $D_{30}$ (as 30T14)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 9*x^29 + 51*x^28 - 225*x^27 + 938*x^26 - 3065*x^25 + 9027*x^24 - 23897*x^23 + 55855*x^22 - 110855*x^21 + 206236*x^20 - 276419*x^19 + 363187*x^18 - 228654*x^17 - 26451*x^16 + 1021780*x^15 - 676090*x^14 + 4072978*x^13 + 297137*x^12 + 7819653*x^11 - 19045735*x^10 + 54647605*x^9 - 99924364*x^8 + 139085676*x^7 - 209111579*x^6 + 87062190*x^5 - 13968112*x^4 - 6858710*x^3 - 581573*x^2 - 428582*x - 60395)
 
gp: K = bnfinit(y^30 - 9*y^29 + 51*y^28 - 225*y^27 + 938*y^26 - 3065*y^25 + 9027*y^24 - 23897*y^23 + 55855*y^22 - 110855*y^21 + 206236*y^20 - 276419*y^19 + 363187*y^18 - 228654*y^17 - 26451*y^16 + 1021780*y^15 - 676090*y^14 + 4072978*y^13 + 297137*y^12 + 7819653*y^11 - 19045735*y^10 + 54647605*y^9 - 99924364*y^8 + 139085676*y^7 - 209111579*y^6 + 87062190*y^5 - 13968112*y^4 - 6858710*y^3 - 581573*y^2 - 428582*y - 60395, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 9*x^29 + 51*x^28 - 225*x^27 + 938*x^26 - 3065*x^25 + 9027*x^24 - 23897*x^23 + 55855*x^22 - 110855*x^21 + 206236*x^20 - 276419*x^19 + 363187*x^18 - 228654*x^17 - 26451*x^16 + 1021780*x^15 - 676090*x^14 + 4072978*x^13 + 297137*x^12 + 7819653*x^11 - 19045735*x^10 + 54647605*x^9 - 99924364*x^8 + 139085676*x^7 - 209111579*x^6 + 87062190*x^5 - 13968112*x^4 - 6858710*x^3 - 581573*x^2 - 428582*x - 60395);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 9*x^29 + 51*x^28 - 225*x^27 + 938*x^26 - 3065*x^25 + 9027*x^24 - 23897*x^23 + 55855*x^22 - 110855*x^21 + 206236*x^20 - 276419*x^19 + 363187*x^18 - 228654*x^17 - 26451*x^16 + 1021780*x^15 - 676090*x^14 + 4072978*x^13 + 297137*x^12 + 7819653*x^11 - 19045735*x^10 + 54647605*x^9 - 99924364*x^8 + 139085676*x^7 - 209111579*x^6 + 87062190*x^5 - 13968112*x^4 - 6858710*x^3 - 581573*x^2 - 428582*x - 60395)
 

\( x^{30} - 9 x^{29} + 51 x^{28} - 225 x^{27} + 938 x^{26} - 3065 x^{25} + 9027 x^{24} - 23897 x^{23} + \cdots - 60395 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[2, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(4939009167239391016458724484733673047883402323239181\) \(\medspace = 3^{15}\cdot 7^{25}\cdot 47^{14}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(52.86\)
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:  $3^{1/2}7^{5/6}47^{1/2}\approx 60.09770990472659$
Ramified primes:   \(3\), \(7\), \(47\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{21}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{5}a^{17}-\frac{2}{5}a^{16}-\frac{2}{5}a^{14}-\frac{1}{5}a^{13}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{2}{5}a^{2}-\frac{1}{5}a$, $\frac{1}{5}a^{18}+\frac{1}{5}a^{16}-\frac{2}{5}a^{15}-\frac{2}{5}a^{13}+\frac{1}{5}a^{6}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}-\frac{2}{5}a$, $\frac{1}{5}a^{19}+\frac{1}{5}a^{13}+\frac{1}{5}a^{7}+\frac{1}{5}a$, $\frac{1}{5}a^{20}+\frac{1}{5}a^{14}+\frac{1}{5}a^{8}+\frac{1}{5}a^{2}$, $\frac{1}{5}a^{21}+\frac{1}{5}a^{15}+\frac{1}{5}a^{9}+\frac{1}{5}a^{3}$, $\frac{1}{5}a^{22}+\frac{1}{5}a^{16}+\frac{1}{5}a^{10}+\frac{1}{5}a^{4}$, $\frac{1}{25}a^{23}-\frac{2}{25}a^{22}-\frac{1}{25}a^{21}+\frac{1}{25}a^{19}-\frac{2}{25}a^{18}+\frac{3}{25}a^{16}-\frac{7}{25}a^{15}+\frac{7}{25}a^{14}+\frac{1}{25}a^{13}-\frac{1}{5}a^{12}+\frac{6}{25}a^{11}+\frac{8}{25}a^{10}-\frac{1}{25}a^{9}+\frac{2}{5}a^{8}+\frac{11}{25}a^{7}+\frac{3}{25}a^{6}-\frac{1}{5}a^{5}+\frac{8}{25}a^{4}+\frac{8}{25}a^{3}+\frac{12}{25}a^{2}+\frac{1}{25}a+\frac{2}{5}$, $\frac{1}{275}a^{24}-\frac{2}{275}a^{23}+\frac{24}{275}a^{22}-\frac{3}{55}a^{21}-\frac{14}{275}a^{20}-\frac{27}{275}a^{19}-\frac{1}{55}a^{18}+\frac{13}{275}a^{17}+\frac{93}{275}a^{16}+\frac{2}{275}a^{15}-\frac{9}{275}a^{14}+\frac{19}{55}a^{13}-\frac{19}{275}a^{12}+\frac{3}{25}a^{11}-\frac{26}{275}a^{10}-\frac{21}{55}a^{9}-\frac{4}{275}a^{8}-\frac{97}{275}a^{7}-\frac{2}{55}a^{6}-\frac{7}{275}a^{5}-\frac{17}{275}a^{4}+\frac{7}{275}a^{3}+\frac{91}{275}a^{2}+\frac{2}{5}a+\frac{1}{11}$, $\frac{1}{1375}a^{25}-\frac{2}{1375}a^{24}-\frac{4}{275}a^{23}+\frac{73}{1375}a^{22}+\frac{17}{275}a^{21}-\frac{82}{1375}a^{20}-\frac{104}{1375}a^{19}-\frac{64}{1375}a^{18}-\frac{127}{1375}a^{17}-\frac{26}{275}a^{16}+\frac{684}{1375}a^{15}-\frac{378}{1375}a^{14}-\frac{393}{1375}a^{13}+\frac{48}{125}a^{12}+\frac{52}{275}a^{11}+\frac{643}{1375}a^{10}-\frac{91}{275}a^{9}+\frac{233}{1375}a^{8}+\frac{1}{1375}a^{7}+\frac{521}{1375}a^{6}+\frac{258}{1375}a^{5}-\frac{124}{275}a^{4}-\frac{151}{1375}a^{3}-\frac{28}{125}a^{2}-\frac{74}{1375}a-\frac{8}{25}$, $\frac{1}{64625}a^{26}+\frac{2}{64625}a^{25}+\frac{82}{64625}a^{24}-\frac{502}{64625}a^{23}+\frac{4942}{64625}a^{22}+\frac{1633}{64625}a^{21}-\frac{1147}{64625}a^{20}-\frac{39}{2585}a^{19}-\frac{1758}{64625}a^{18}+\frac{22}{5875}a^{17}+\frac{227}{1375}a^{16}-\frac{21622}{64625}a^{15}+\frac{4646}{12925}a^{14}-\frac{23319}{64625}a^{13}+\frac{15407}{64625}a^{12}-\frac{1542}{5875}a^{11}-\frac{22193}{64625}a^{10}-\frac{32112}{64625}a^{9}-\frac{24807}{64625}a^{8}+\frac{2316}{12925}a^{7}+\frac{19667}{64625}a^{6}-\frac{25658}{64625}a^{5}+\frac{30149}{64625}a^{4}+\frac{19658}{64625}a^{3}-\frac{20171}{64625}a^{2}-\frac{8986}{64625}a-\frac{6}{25}$, $\frac{1}{323125}a^{27}-\frac{2}{323125}a^{26}-\frac{114}{323125}a^{25}+\frac{486}{323125}a^{24}-\frac{302}{64625}a^{23}-\frac{27394}{323125}a^{22}+\frac{24281}{323125}a^{21}+\frac{18794}{323125}a^{20}+\frac{11824}{323125}a^{19}+\frac{22361}{323125}a^{18}+\frac{19947}{323125}a^{17}+\frac{145792}{323125}a^{16}-\frac{125564}{323125}a^{15}-\frac{25203}{64625}a^{14}+\frac{80577}{323125}a^{13}+\frac{49861}{323125}a^{12}+\frac{19001}{64625}a^{11}-\frac{145534}{323125}a^{10}-\frac{41354}{323125}a^{9}-\frac{156481}{323125}a^{8}-\frac{141286}{323125}a^{7}-\frac{126369}{323125}a^{6}-\frac{90328}{323125}a^{5}-\frac{147703}{323125}a^{4}+\frac{24974}{64625}a^{3}+\frac{26437}{323125}a^{2}+\frac{140331}{323125}a+\frac{1}{1375}$, $\frac{1}{337665625}a^{28}-\frac{7}{7184375}a^{27}-\frac{87}{13506625}a^{26}+\frac{311}{1615625}a^{25}-\frac{179317}{337665625}a^{24}-\frac{4170244}{337665625}a^{23}+\frac{24621634}{337665625}a^{22}+\frac{14394187}{337665625}a^{21}-\frac{11856414}{337665625}a^{20}+\frac{183987}{17771875}a^{19}-\frac{4205123}{67533125}a^{18}-\frac{20350887}{337665625}a^{17}-\frac{115068533}{337665625}a^{16}-\frac{99223147}{337665625}a^{15}-\frac{101873963}{337665625}a^{14}-\frac{127490553}{337665625}a^{13}+\frac{84443773}{337665625}a^{12}+\frac{2557944}{17771875}a^{11}-\frac{150985146}{337665625}a^{10}+\frac{96429632}{337665625}a^{9}-\frac{108770349}{337665625}a^{8}+\frac{57625118}{337665625}a^{7}-\frac{7324756}{67533125}a^{6}-\frac{87245932}{337665625}a^{5}-\frac{122636784}{337665625}a^{4}-\frac{136466713}{337665625}a^{3}-\frac{89749373}{337665625}a^{2}+\frac{126060553}{337665625}a-\frac{463897}{1436875}$, $\frac{1}{38\!\cdots\!75}a^{29}+\frac{18\!\cdots\!78}{38\!\cdots\!75}a^{28}-\frac{55\!\cdots\!33}{38\!\cdots\!75}a^{27}+\frac{27\!\cdots\!84}{38\!\cdots\!75}a^{26}+\frac{51\!\cdots\!21}{38\!\cdots\!75}a^{25}+\frac{66\!\cdots\!57}{38\!\cdots\!75}a^{24}-\frac{73\!\cdots\!49}{38\!\cdots\!75}a^{23}-\frac{57\!\cdots\!01}{77\!\cdots\!75}a^{22}+\frac{99\!\cdots\!58}{77\!\cdots\!75}a^{21}+\frac{46\!\cdots\!72}{77\!\cdots\!75}a^{20}-\frac{41\!\cdots\!14}{38\!\cdots\!75}a^{19}-\frac{38\!\cdots\!72}{38\!\cdots\!75}a^{18}-\frac{18\!\cdots\!02}{38\!\cdots\!75}a^{17}+\frac{51\!\cdots\!12}{38\!\cdots\!75}a^{16}+\frac{86\!\cdots\!03}{38\!\cdots\!75}a^{15}-\frac{14\!\cdots\!26}{20\!\cdots\!25}a^{14}+\frac{82\!\cdots\!22}{35\!\cdots\!25}a^{13}+\frac{15\!\cdots\!92}{38\!\cdots\!75}a^{12}-\frac{17\!\cdots\!94}{38\!\cdots\!75}a^{11}-\frac{27\!\cdots\!74}{70\!\cdots\!25}a^{10}+\frac{22\!\cdots\!54}{77\!\cdots\!75}a^{9}-\frac{21\!\cdots\!39}{77\!\cdots\!75}a^{8}+\frac{17\!\cdots\!41}{35\!\cdots\!25}a^{7}-\frac{78\!\cdots\!66}{57\!\cdots\!25}a^{6}-\frac{24\!\cdots\!18}{38\!\cdots\!75}a^{5}+\frac{99\!\cdots\!84}{39\!\cdots\!25}a^{4}-\frac{81\!\cdots\!78}{29\!\cdots\!75}a^{3}+\frac{29\!\cdots\!84}{16\!\cdots\!25}a^{2}+\frac{70\!\cdots\!46}{38\!\cdots\!75}a-\frac{23\!\cdots\!84}{16\!\cdots\!25}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $5$

Class group and class number

not computed

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:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^30 - 9*x^29 + 51*x^28 - 225*x^27 + 938*x^26 - 3065*x^25 + 9027*x^24 - 23897*x^23 + 55855*x^22 - 110855*x^21 + 206236*x^20 - 276419*x^19 + 363187*x^18 - 228654*x^17 - 26451*x^16 + 1021780*x^15 - 676090*x^14 + 4072978*x^13 + 297137*x^12 + 7819653*x^11 - 19045735*x^10 + 54647605*x^9 - 99924364*x^8 + 139085676*x^7 - 209111579*x^6 + 87062190*x^5 - 13968112*x^4 - 6858710*x^3 - 581573*x^2 - 428582*x - 60395)
 
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^30 - 9*x^29 + 51*x^28 - 225*x^27 + 938*x^26 - 3065*x^25 + 9027*x^24 - 23897*x^23 + 55855*x^22 - 110855*x^21 + 206236*x^20 - 276419*x^19 + 363187*x^18 - 228654*x^17 - 26451*x^16 + 1021780*x^15 - 676090*x^14 + 4072978*x^13 + 297137*x^12 + 7819653*x^11 - 19045735*x^10 + 54647605*x^9 - 99924364*x^8 + 139085676*x^7 - 209111579*x^6 + 87062190*x^5 - 13968112*x^4 - 6858710*x^3 - 581573*x^2 - 428582*x - 60395, 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^30 - 9*x^29 + 51*x^28 - 225*x^27 + 938*x^26 - 3065*x^25 + 9027*x^24 - 23897*x^23 + 55855*x^22 - 110855*x^21 + 206236*x^20 - 276419*x^19 + 363187*x^18 - 228654*x^17 - 26451*x^16 + 1021780*x^15 - 676090*x^14 + 4072978*x^13 + 297137*x^12 + 7819653*x^11 - 19045735*x^10 + 54647605*x^9 - 99924364*x^8 + 139085676*x^7 - 209111579*x^6 + 87062190*x^5 - 13968112*x^4 - 6858710*x^3 - 581573*x^2 - 428582*x - 60395);
 
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^30 - 9*x^29 + 51*x^28 - 225*x^27 + 938*x^26 - 3065*x^25 + 9027*x^24 - 23897*x^23 + 55855*x^22 - 110855*x^21 + 206236*x^20 - 276419*x^19 + 363187*x^18 - 228654*x^17 - 26451*x^16 + 1021780*x^15 - 676090*x^14 + 4072978*x^13 + 297137*x^12 + 7819653*x^11 - 19045735*x^10 + 54647605*x^9 - 99924364*x^8 + 139085676*x^7 - 209111579*x^6 + 87062190*x^5 - 13968112*x^4 - 6858710*x^3 - 581573*x^2 - 428582*x - 60395);
 
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

$D_{30}$ (as 30T14):

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 60
The 18 conjugacy class representatives for $D_{30}$
Character table for $D_{30}$

Intermediate fields

\(\Q(\sqrt{21}) \), 3.1.2303.1, 5.1.2209.1, 6.2.1002419901.1, 10.2.19929110051781.1, 15.1.143108492101942920287.1

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

Degree 30 sibling: 30.0.232133430860251377773560050782482633250519909192241507.1
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 $30$ R ${\href{/padicField/5.2.0.1}{2} }^{14}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ R ${\href{/padicField/11.2.0.1}{2} }^{15}$ ${\href{/padicField/13.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{15}$ ${\href{/padicField/23.2.0.1}{2} }^{15}$ ${\href{/padicField/29.2.0.1}{2} }^{15}$ ${\href{/padicField/31.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{14}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{14}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ R $30$ $15^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.10.5.1$x^{10} + 162 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} + 162 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} + 162 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(7\) Copy content Toggle raw display Deg $30$$6$$5$$25$
\(47\) Copy content Toggle raw display $\Q_{47}$$x + 42$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 42$$1$$1$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.987.2t1.a.a$1$ $ 3 \cdot 7 \cdot 47 $ \(\Q(\sqrt{-987}) \) $C_2$ (as 2T1) $1$ $-1$
1.47.2t1.a.a$1$ $ 47 $ \(\Q(\sqrt{-47}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.21.2t1.a.a$1$ $ 3 \cdot 7 $ \(\Q(\sqrt{21}) \) $C_2$ (as 2T1) $1$ $1$
* 2.20727.6t3.a.a$2$ $ 3^{2} \cdot 7^{2} \cdot 47 $ 6.0.47113735347.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.2303.3t2.a.a$2$ $ 7^{2} \cdot 47 $ 3.1.2303.1 $S_3$ (as 3T2) $1$ $0$
* 2.47.5t2.a.b$2$ $ 47 $ 5.1.2209.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.47.5t2.a.a$2$ $ 47 $ 5.1.2209.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.20727.10t3.b.b$2$ $ 3^{2} \cdot 7^{2} \cdot 47 $ 10.0.936668172433707.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.20727.10t3.b.a$2$ $ 3^{2} \cdot 7^{2} \cdot 47 $ 10.0.936668172433707.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.20727.30t14.b.c$2$ $ 3^{2} \cdot 7^{2} \cdot 47 $ 30.2.4939009167239391016458724484733673047883402323239181.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2303.15t2.a.a$2$ $ 7^{2} \cdot 47 $ 15.1.143108492101942920287.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2303.15t2.a.c$2$ $ 7^{2} \cdot 47 $ 15.1.143108492101942920287.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.20727.30t14.b.a$2$ $ 3^{2} \cdot 7^{2} \cdot 47 $ 30.2.4939009167239391016458724484733673047883402323239181.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2303.15t2.a.b$2$ $ 7^{2} \cdot 47 $ 15.1.143108492101942920287.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.20727.30t14.b.b$2$ $ 3^{2} \cdot 7^{2} \cdot 47 $ 30.2.4939009167239391016458724484733673047883402323239181.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2303.15t2.a.d$2$ $ 7^{2} \cdot 47 $ 15.1.143108492101942920287.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.20727.30t14.b.d$2$ $ 3^{2} \cdot 7^{2} \cdot 47 $ 30.2.4939009167239391016458724484733673047883402323239181.1 $D_{30}$ (as 30T14) $1$ $0$

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 *.