Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 11*x^16 + 24*x^15 - 121*x^14 + 36*x^13 + 803*x^12 - 1458*x^11 - 1679*x^10 + 8232*x^9 - 4428*x^8 - 10054*x^7 + 3009*x^6 + 22712*x^5 + 79531*x^4 - 343084*x^3 + 511207*x^2 - 353486*x + 92561)
gp: K = bnfinit(y^18 - 6*y^17 + 11*y^16 + 24*y^15 - 121*y^14 + 36*y^13 + 803*y^12 - 1458*y^11 - 1679*y^10 + 8232*y^9 - 4428*y^8 - 10054*y^7 + 3009*y^6 + 22712*y^5 + 79531*y^4 - 343084*y^3 + 511207*y^2 - 353486*y + 92561, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 11*x^16 + 24*x^15 - 121*x^14 + 36*x^13 + 803*x^12 - 1458*x^11 - 1679*x^10 + 8232*x^9 - 4428*x^8 - 10054*x^7 + 3009*x^6 + 22712*x^5 + 79531*x^4 - 343084*x^3 + 511207*x^2 - 353486*x + 92561);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 6*x^17 + 11*x^16 + 24*x^15 - 121*x^14 + 36*x^13 + 803*x^12 - 1458*x^11 - 1679*x^10 + 8232*x^9 - 4428*x^8 - 10054*x^7 + 3009*x^6 + 22712*x^5 + 79531*x^4 - 343084*x^3 + 511207*x^2 - 353486*x + 92561)
\( x^{18} - 6 x^{17} + 11 x^{16} + 24 x^{15} - 121 x^{14} + 36 x^{13} + 803 x^{12} - 1458 x^{11} + \cdots + 92561 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-580207226143679709184000000000\)
\(\medspace = -\,2^{27}\cdot 5^{9}\cdot 19^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(45.03\) | 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: | $2^{3/2}5^{1/2}19^{2/3}\approx 45.0331572624958$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-10}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{21}a^{12}+\frac{10}{21}a^{11}+\frac{2}{7}a^{10}+\frac{1}{21}a^{9}+\frac{1}{3}a^{8}-\frac{8}{21}a^{7}+\frac{2}{21}a^{6}+\frac{1}{3}a^{5}+\frac{1}{7}a^{4}+\frac{8}{21}a^{2}-\frac{4}{21}a+\frac{1}{3}$, $\frac{1}{21}a^{13}-\frac{10}{21}a^{11}+\frac{4}{21}a^{10}-\frac{1}{7}a^{9}+\frac{2}{7}a^{8}-\frac{2}{21}a^{7}+\frac{8}{21}a^{6}-\frac{4}{21}a^{5}-\frac{3}{7}a^{4}+\frac{8}{21}a^{3}+\frac{5}{21}a-\frac{1}{3}$, $\frac{1}{21}a^{14}-\frac{1}{21}a^{11}-\frac{2}{7}a^{10}-\frac{5}{21}a^{9}+\frac{5}{21}a^{8}-\frac{3}{7}a^{7}-\frac{5}{21}a^{6}-\frac{2}{21}a^{5}-\frac{4}{21}a^{4}+\frac{1}{21}a^{2}-\frac{5}{21}a+\frac{1}{3}$, $\frac{1}{21}a^{15}+\frac{4}{21}a^{11}+\frac{1}{21}a^{10}+\frac{2}{7}a^{9}-\frac{2}{21}a^{8}+\frac{8}{21}a^{7}+\frac{1}{7}a^{5}+\frac{1}{7}a^{4}+\frac{1}{21}a^{3}+\frac{1}{7}a^{2}+\frac{1}{7}a+\frac{1}{3}$, $\frac{1}{21}a^{16}+\frac{1}{7}a^{11}+\frac{1}{7}a^{10}-\frac{2}{7}a^{9}+\frac{1}{21}a^{8}-\frac{10}{21}a^{7}-\frac{5}{21}a^{6}-\frac{4}{21}a^{5}+\frac{10}{21}a^{4}+\frac{1}{7}a^{3}-\frac{8}{21}a^{2}+\frac{2}{21}a-\frac{1}{3}$, $\frac{1}{25\!\cdots\!43}a^{17}-\frac{17\!\cdots\!40}{85\!\cdots\!81}a^{16}+\frac{12\!\cdots\!70}{85\!\cdots\!81}a^{15}+\frac{13\!\cdots\!80}{25\!\cdots\!43}a^{14}+\frac{59\!\cdots\!70}{25\!\cdots\!43}a^{13}-\frac{92\!\cdots\!63}{25\!\cdots\!43}a^{12}-\frac{32\!\cdots\!65}{85\!\cdots\!81}a^{11}-\frac{21\!\cdots\!42}{52\!\cdots\!07}a^{10}-\frac{12\!\cdots\!44}{25\!\cdots\!43}a^{9}-\frac{32\!\cdots\!85}{85\!\cdots\!81}a^{8}-\frac{11\!\cdots\!67}{36\!\cdots\!49}a^{7}-\frac{27\!\cdots\!33}{85\!\cdots\!81}a^{6}+\frac{85\!\cdots\!61}{85\!\cdots\!81}a^{5}+\frac{78\!\cdots\!92}{25\!\cdots\!43}a^{4}+\frac{28\!\cdots\!10}{85\!\cdots\!81}a^{3}-\frac{11\!\cdots\!00}{25\!\cdots\!43}a^{2}+\frac{26\!\cdots\!68}{17\!\cdots\!69}a+\frac{17\!\cdots\!73}{52\!\cdots\!07}$
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
$C_{78}$, which has order $78$ (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: | $8$ | 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: |
$\frac{57\!\cdots\!30}{25\!\cdots\!43}a^{17}-\frac{29\!\cdots\!27}{25\!\cdots\!43}a^{16}+\frac{39\!\cdots\!87}{25\!\cdots\!43}a^{15}+\frac{56\!\cdots\!92}{85\!\cdots\!81}a^{14}-\frac{18\!\cdots\!52}{85\!\cdots\!81}a^{13}-\frac{23\!\cdots\!97}{25\!\cdots\!43}a^{12}+\frac{14\!\cdots\!94}{85\!\cdots\!81}a^{11}-\frac{69\!\cdots\!04}{36\!\cdots\!49}a^{10}-\frac{44\!\cdots\!76}{85\!\cdots\!81}a^{9}+\frac{36\!\cdots\!49}{25\!\cdots\!43}a^{8}+\frac{16\!\cdots\!51}{12\!\cdots\!83}a^{7}-\frac{55\!\cdots\!39}{25\!\cdots\!43}a^{6}-\frac{26\!\cdots\!45}{25\!\cdots\!43}a^{5}+\frac{10\!\cdots\!91}{25\!\cdots\!43}a^{4}+\frac{54\!\cdots\!38}{25\!\cdots\!43}a^{3}-\frac{15\!\cdots\!66}{25\!\cdots\!43}a^{2}+\frac{35\!\cdots\!46}{52\!\cdots\!07}a-\frac{13\!\cdots\!49}{52\!\cdots\!07}$, $\frac{62\!\cdots\!44}{25\!\cdots\!43}a^{17}-\frac{32\!\cdots\!64}{25\!\cdots\!43}a^{16}+\frac{44\!\cdots\!86}{25\!\cdots\!43}a^{15}+\frac{60\!\cdots\!66}{85\!\cdots\!81}a^{14}-\frac{20\!\cdots\!51}{85\!\cdots\!81}a^{13}-\frac{23\!\cdots\!10}{25\!\cdots\!43}a^{12}+\frac{16\!\cdots\!18}{85\!\cdots\!81}a^{11}-\frac{78\!\cdots\!39}{36\!\cdots\!49}a^{10}-\frac{48\!\cdots\!71}{85\!\cdots\!81}a^{9}+\frac{40\!\cdots\!03}{25\!\cdots\!43}a^{8}+\frac{13\!\cdots\!57}{12\!\cdots\!83}a^{7}-\frac{61\!\cdots\!37}{25\!\cdots\!43}a^{6}-\frac{26\!\cdots\!03}{25\!\cdots\!43}a^{5}+\frac{12\!\cdots\!71}{25\!\cdots\!43}a^{4}+\frac{58\!\cdots\!50}{25\!\cdots\!43}a^{3}-\frac{17\!\cdots\!55}{25\!\cdots\!43}a^{2}+\frac{39\!\cdots\!20}{52\!\cdots\!07}a-\frac{16\!\cdots\!00}{52\!\cdots\!07}$, $\frac{25\!\cdots\!64}{69\!\cdots\!39}a^{17}-\frac{17\!\cdots\!24}{69\!\cdots\!39}a^{16}+\frac{35\!\cdots\!13}{69\!\cdots\!39}a^{15}+\frac{21\!\cdots\!22}{23\!\cdots\!13}a^{14}-\frac{12\!\cdots\!56}{23\!\cdots\!13}a^{13}+\frac{19\!\cdots\!99}{69\!\cdots\!39}a^{12}+\frac{81\!\cdots\!56}{23\!\cdots\!13}a^{11}-\frac{72\!\cdots\!26}{99\!\cdots\!77}a^{10}-\frac{16\!\cdots\!86}{23\!\cdots\!13}a^{9}+\frac{28\!\cdots\!41}{69\!\cdots\!39}a^{8}-\frac{72\!\cdots\!57}{33\!\cdots\!59}a^{7}-\frac{45\!\cdots\!13}{69\!\cdots\!39}a^{6}+\frac{18\!\cdots\!75}{69\!\cdots\!39}a^{5}+\frac{10\!\cdots\!35}{69\!\cdots\!39}a^{4}+\frac{20\!\cdots\!58}{69\!\cdots\!39}a^{3}-\frac{11\!\cdots\!73}{69\!\cdots\!39}a^{2}+\frac{22\!\cdots\!26}{99\!\cdots\!77}a-\frac{13\!\cdots\!51}{14\!\cdots\!11}$, $\frac{94\!\cdots\!66}{85\!\cdots\!81}a^{17}-\frac{49\!\cdots\!87}{85\!\cdots\!81}a^{16}+\frac{66\!\cdots\!83}{85\!\cdots\!81}a^{15}+\frac{27\!\cdots\!05}{85\!\cdots\!81}a^{14}-\frac{93\!\cdots\!97}{85\!\cdots\!81}a^{13}-\frac{11\!\cdots\!44}{25\!\cdots\!43}a^{12}+\frac{22\!\cdots\!15}{25\!\cdots\!43}a^{11}-\frac{11\!\cdots\!61}{12\!\cdots\!83}a^{10}-\frac{66\!\cdots\!86}{25\!\cdots\!43}a^{9}+\frac{18\!\cdots\!50}{25\!\cdots\!43}a^{8}+\frac{21\!\cdots\!23}{36\!\cdots\!49}a^{7}-\frac{27\!\cdots\!96}{25\!\cdots\!43}a^{6}-\frac{12\!\cdots\!00}{25\!\cdots\!43}a^{5}+\frac{18\!\cdots\!49}{85\!\cdots\!81}a^{4}+\frac{89\!\cdots\!48}{85\!\cdots\!81}a^{3}-\frac{77\!\cdots\!24}{25\!\cdots\!43}a^{2}+\frac{12\!\cdots\!59}{36\!\cdots\!49}a-\frac{70\!\cdots\!77}{52\!\cdots\!07}$, $\frac{21\!\cdots\!90}{25\!\cdots\!43}a^{17}-\frac{10\!\cdots\!10}{25\!\cdots\!43}a^{16}+\frac{88\!\cdots\!52}{25\!\cdots\!43}a^{15}+\frac{22\!\cdots\!20}{85\!\cdots\!81}a^{14}-\frac{55\!\cdots\!70}{85\!\cdots\!81}a^{13}-\frac{19\!\cdots\!60}{25\!\cdots\!43}a^{12}+\frac{15\!\cdots\!31}{25\!\cdots\!43}a^{11}-\frac{12\!\cdots\!49}{36\!\cdots\!49}a^{10}-\frac{54\!\cdots\!57}{25\!\cdots\!43}a^{9}+\frac{32\!\cdots\!02}{85\!\cdots\!81}a^{8}+\frac{13\!\cdots\!43}{52\!\cdots\!07}a^{7}-\frac{40\!\cdots\!81}{85\!\cdots\!81}a^{6}-\frac{73\!\cdots\!66}{85\!\cdots\!81}a^{5}+\frac{15\!\cdots\!24}{25\!\cdots\!43}a^{4}+\frac{22\!\cdots\!22}{25\!\cdots\!43}a^{3}-\frac{14\!\cdots\!57}{85\!\cdots\!81}a^{2}+\frac{52\!\cdots\!81}{36\!\cdots\!49}a-\frac{99\!\cdots\!33}{17\!\cdots\!69}$, $\frac{27\!\cdots\!02}{25\!\cdots\!43}a^{17}-\frac{47\!\cdots\!90}{85\!\cdots\!81}a^{16}+\frac{60\!\cdots\!02}{85\!\cdots\!81}a^{15}+\frac{81\!\cdots\!39}{25\!\cdots\!43}a^{14}-\frac{26\!\cdots\!37}{25\!\cdots\!43}a^{13}-\frac{41\!\cdots\!60}{85\!\cdots\!81}a^{12}+\frac{21\!\cdots\!71}{25\!\cdots\!43}a^{11}-\frac{45\!\cdots\!47}{52\!\cdots\!07}a^{10}-\frac{21\!\cdots\!16}{85\!\cdots\!81}a^{9}+\frac{57\!\cdots\!60}{85\!\cdots\!81}a^{8}+\frac{16\!\cdots\!16}{17\!\cdots\!69}a^{7}-\frac{25\!\cdots\!13}{25\!\cdots\!43}a^{6}-\frac{13\!\cdots\!99}{25\!\cdots\!43}a^{5}+\frac{51\!\cdots\!14}{25\!\cdots\!43}a^{4}+\frac{26\!\cdots\!05}{25\!\cdots\!43}a^{3}-\frac{72\!\cdots\!18}{25\!\cdots\!43}a^{2}+\frac{11\!\cdots\!27}{36\!\cdots\!49}a-\frac{20\!\cdots\!43}{17\!\cdots\!69}$, $\frac{61\!\cdots\!59}{25\!\cdots\!43}a^{17}-\frac{32\!\cdots\!49}{25\!\cdots\!43}a^{16}+\frac{43\!\cdots\!76}{25\!\cdots\!43}a^{15}+\frac{59\!\cdots\!50}{85\!\cdots\!81}a^{14}-\frac{60\!\cdots\!94}{25\!\cdots\!43}a^{13}-\frac{22\!\cdots\!25}{25\!\cdots\!43}a^{12}+\frac{47\!\cdots\!10}{25\!\cdots\!43}a^{11}-\frac{77\!\cdots\!42}{36\!\cdots\!49}a^{10}-\frac{47\!\cdots\!95}{85\!\cdots\!81}a^{9}+\frac{39\!\cdots\!36}{25\!\cdots\!43}a^{8}+\frac{25\!\cdots\!48}{36\!\cdots\!49}a^{7}-\frac{60\!\cdots\!55}{25\!\cdots\!43}a^{6}-\frac{78\!\cdots\!50}{85\!\cdots\!81}a^{5}+\frac{12\!\cdots\!10}{25\!\cdots\!43}a^{4}+\frac{57\!\cdots\!96}{25\!\cdots\!43}a^{3}-\frac{16\!\cdots\!21}{25\!\cdots\!43}a^{2}+\frac{90\!\cdots\!55}{12\!\cdots\!83}a-\frac{51\!\cdots\!11}{17\!\cdots\!69}$, $\frac{12\!\cdots\!42}{32\!\cdots\!93}a^{17}-\frac{20\!\cdots\!52}{96\!\cdots\!79}a^{16}+\frac{91\!\cdots\!05}{32\!\cdots\!93}a^{15}+\frac{11\!\cdots\!56}{96\!\cdots\!79}a^{14}-\frac{38\!\cdots\!12}{96\!\cdots\!79}a^{13}-\frac{14\!\cdots\!02}{96\!\cdots\!79}a^{12}+\frac{99\!\cdots\!71}{32\!\cdots\!93}a^{11}-\frac{49\!\cdots\!90}{13\!\cdots\!97}a^{10}-\frac{29\!\cdots\!09}{32\!\cdots\!93}a^{9}+\frac{83\!\cdots\!00}{32\!\cdots\!93}a^{8}+\frac{17\!\cdots\!96}{13\!\cdots\!97}a^{7}-\frac{12\!\cdots\!47}{32\!\cdots\!93}a^{6}-\frac{52\!\cdots\!81}{32\!\cdots\!93}a^{5}+\frac{75\!\cdots\!47}{96\!\cdots\!79}a^{4}+\frac{36\!\cdots\!60}{96\!\cdots\!79}a^{3}-\frac{10\!\cdots\!00}{96\!\cdots\!79}a^{2}+\frac{17\!\cdots\!81}{13\!\cdots\!97}a-\frac{33\!\cdots\!16}{65\!\cdots\!57}$
| 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: | \( 633916.9051977703 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 633916.9051977703 \cdot 78}{2\cdot\sqrt{580207226143679709184000000000}}\cr\approx \mathstrut & 0.495363712140735 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 11*x^16 + 24*x^15 - 121*x^14 + 36*x^13 + 803*x^12 - 1458*x^11 - 1679*x^10 + 8232*x^9 - 4428*x^8 - 10054*x^7 + 3009*x^6 + 22712*x^5 + 79531*x^4 - 343084*x^3 + 511207*x^2 - 353486*x + 92561)
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^18 - 6*x^17 + 11*x^16 + 24*x^15 - 121*x^14 + 36*x^13 + 803*x^12 - 1458*x^11 - 1679*x^10 + 8232*x^9 - 4428*x^8 - 10054*x^7 + 3009*x^6 + 22712*x^5 + 79531*x^4 - 343084*x^3 + 511207*x^2 - 353486*x + 92561, 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^18 - 6*x^17 + 11*x^16 + 24*x^15 - 121*x^14 + 36*x^13 + 803*x^12 - 1458*x^11 - 1679*x^10 + 8232*x^9 - 4428*x^8 - 10054*x^7 + 3009*x^6 + 22712*x^5 + 79531*x^4 - 343084*x^3 + 511207*x^2 - 353486*x + 92561);
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^18 - 6*x^17 + 11*x^16 + 24*x^15 - 121*x^14 + 36*x^13 + 803*x^12 - 1458*x^11 - 1679*x^10 + 8232*x^9 - 4428*x^8 - 10054*x^7 + 3009*x^6 + 22712*x^5 + 79531*x^4 - 343084*x^3 + 511207*x^2 - 353486*x + 92561);
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
$C_6\times S_3$ (as 18T6):
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 36 |
| The 18 conjugacy class representatives for $S_3 \times C_6$ |
| Character table for $S_3 \times C_6$ |
Intermediate fields
| \(\Q(\sqrt{-10}) \), 3.1.2888.1, 3.3.361.1, 6.0.8340544000.2, 6.0.8340544000.3, 9.3.24087491072.1 |
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: | data not computed |
| Degree 12 sibling: | 12.0.533794816000000.1 |
| Degree 18 sibling: | 18.6.1133217238561874432000000000.2 |
| Minimal sibling: | 12.0.533794816000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }^{6}$ | ${\href{/padicField/11.3.0.1}{3} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{9}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.9.7 | $x^{6} + 32 x^{4} + 2 x^{3} + 301 x^{2} - 58 x + 811$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ |
| 2.12.18.15 | $x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ | |
|
\(5\)
| 5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(19\)
| 19.9.6.2 | $x^{9} + 981 x^{7} + 108 x^{6} + 316911 x^{5} + 20529 x^{4} + 34115982 x^{3} + 10990188 x^{2} + 130942880 x + 566550143$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| 19.9.6.2 | $x^{9} + 981 x^{7} + 108 x^{6} + 316911 x^{5} + 20529 x^{4} + 34115982 x^{3} + 10990188 x^{2} + 130942880 x + 566550143$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.40.2t1.b.a | $1$ | $ 2^{3} \cdot 5 $ | \(\Q(\sqrt{-10}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.760.6t1.b.a | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.8340544000.3 | $C_6$ (as 6T1) | $0$ | $-1$ |
| 1.95.6t1.a.a | $1$ | $ 5 \cdot 19 $ | 6.6.16290125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.95.6t1.a.b | $1$ | $ 5 \cdot 19 $ | 6.6.16290125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.152.6t1.c.a | $1$ | $ 2^{3} \cdot 19 $ | 6.0.66724352.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.760.6t1.b.b | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.8340544000.3 | $C_6$ (as 6T1) | $0$ | $-1$ |
| 1.152.6t1.c.b | $1$ | $ 2^{3} \cdot 19 $ | 6.0.66724352.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 2.2888.3t2.b.a | $2$ | $ 2^{3} \cdot 19^{2}$ | 3.1.2888.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.72200.6t3.d.a | $2$ | $ 2^{3} \cdot 5^{2} \cdot 19^{2}$ | 6.0.8340544000.2 | $D_{6}$ (as 6T3) | $1$ | $0$ |
| * | 2.152.6t5.a.a | $2$ | $ 2^{3} \cdot 19 $ | 6.0.184832.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.152.6t5.a.b | $2$ | $ 2^{3} \cdot 19 $ | 6.0.184832.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.3800.12t18.g.a | $2$ | $ 2^{3} \cdot 5^{2} \cdot 19 $ | 18.0.580207226143679709184000000000.2 | $S_3 \times C_6$ (as 18T6) | $0$ | $0$ |
| * | 2.3800.12t18.g.b | $2$ | $ 2^{3} \cdot 5^{2} \cdot 19 $ | 18.0.580207226143679709184000000000.2 | $S_3 \times C_6$ (as 18T6) | $0$ | $0$ |
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 *.