Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 17*x^16 - 26*x^15 + 40*x^14 - 98*x^13 + 214*x^12 - 180*x^11 + 94*x^10 + 240*x^9 - 166*x^8 - 346*x^7 + 387*x^6 + 144*x^5 + 183*x^4 + 45*x^3 + 21*x^2 + 3*x + 1)
gp: K = bnfinit(y^18 - 6*y^17 + 17*y^16 - 26*y^15 + 40*y^14 - 98*y^13 + 214*y^12 - 180*y^11 + 94*y^10 + 240*y^9 - 166*y^8 - 346*y^7 + 387*y^6 + 144*y^5 + 183*y^4 + 45*y^3 + 21*y^2 + 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 17*x^16 - 26*x^15 + 40*x^14 - 98*x^13 + 214*x^12 - 180*x^11 + 94*x^10 + 240*x^9 - 166*x^8 - 346*x^7 + 387*x^6 + 144*x^5 + 183*x^4 + 45*x^3 + 21*x^2 + 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 + 17*x^16 - 26*x^15 + 40*x^14 - 98*x^13 + 214*x^12 - 180*x^11 + 94*x^10 + 240*x^9 - 166*x^8 - 346*x^7 + 387*x^6 + 144*x^5 + 183*x^4 + 45*x^3 + 21*x^2 + 3*x + 1)
\( x^{18} - 6 x^{17} + 17 x^{16} - 26 x^{15} + 40 x^{14} - 98 x^{13} + 214 x^{12} - 180 x^{11} + 94 x^{10} + \cdots + 1 \)
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: |
\(-105226667788517176205647\)
\(\medspace = -\,7^{15}\cdot 53^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.01\) | 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: | $7^{5/6}53^{1/2}\approx 36.8456567103712$ | ||
| Ramified primes: |
\(7\), \(53\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{256}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}+\frac{1}{5}a^{11}-\frac{2}{5}a^{10}-\frac{1}{5}a^{9}+\frac{2}{5}a^{8}+\frac{2}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{13}+\frac{2}{5}a^{11}+\frac{1}{5}a^{10}-\frac{2}{5}a^{9}+\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{14}-\frac{1}{5}a^{11}+\frac{2}{5}a^{10}+\frac{2}{5}a^{9}+\frac{2}{5}a^{8}+\frac{2}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}$, $\frac{1}{5}a^{15}-\frac{2}{5}a^{11}+\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{2}{5}$, $\frac{1}{10}a^{16}+\frac{1}{5}a^{11}+\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{7}-\frac{2}{5}a^{6}-\frac{2}{5}a^{5}+\frac{1}{10}a^{4}+\frac{1}{5}a^{3}-\frac{1}{2}a+\frac{1}{10}$, $\frac{1}{98\!\cdots\!60}a^{17}-\frac{47\!\cdots\!81}{98\!\cdots\!60}a^{16}-\frac{12\!\cdots\!47}{49\!\cdots\!13}a^{15}-\frac{24\!\cdots\!61}{49\!\cdots\!30}a^{14}-\frac{32\!\cdots\!61}{49\!\cdots\!30}a^{13}+\frac{19\!\cdots\!47}{24\!\cdots\!65}a^{12}+\frac{48\!\cdots\!47}{49\!\cdots\!30}a^{11}+\frac{33\!\cdots\!01}{49\!\cdots\!30}a^{10}-\frac{38\!\cdots\!37}{24\!\cdots\!65}a^{9}-\frac{22\!\cdots\!39}{24\!\cdots\!65}a^{8}-\frac{18\!\cdots\!77}{49\!\cdots\!30}a^{7}+\frac{19\!\cdots\!47}{24\!\cdots\!65}a^{6}+\frac{31\!\cdots\!71}{98\!\cdots\!60}a^{5}-\frac{11\!\cdots\!41}{98\!\cdots\!60}a^{4}-\frac{14\!\cdots\!11}{49\!\cdots\!30}a^{3}-\frac{17\!\cdots\!61}{98\!\cdots\!60}a^{2}-\frac{48\!\cdots\!13}{24\!\cdots\!65}a+\frac{81\!\cdots\!51}{19\!\cdots\!52}$
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
$C_{2}$, which has order $2$
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: |
\( -\frac{1079221961855783189}{24686269706510276065} a^{17} + \frac{9559957616114906433}{49372539413020552130} a^{16} - \frac{7862821012445822079}{24686269706510276065} a^{15} - \frac{2698281314406409891}{24686269706510276065} a^{14} + \frac{1334614879615159058}{4937253941302055213} a^{13} + \frac{28353905884660856891}{24686269706510276065} a^{12} - \frac{49707137782911400077}{24686269706510276065} a^{11} - \frac{40555873751918850873}{4937253941302055213} a^{10} + \frac{278279393436072230658}{24686269706510276065} a^{9} - \frac{493227439224437268034}{24686269706510276065} a^{8} - \frac{176264853243586564903}{24686269706510276065} a^{7} + \frac{144019997988855549745}{4937253941302055213} a^{6} + \frac{96652775618478636262}{24686269706510276065} a^{5} - \frac{1774626336418581316847}{49372539413020552130} a^{4} - \frac{301758603604306121189}{24686269706510276065} a^{3} - \frac{391478876759448471347}{24686269706510276065} a^{2} - \frac{11110056191179123083}{9874507882604110426} a - \frac{3586954569388599281}{9874507882604110426} \)
(order $14$)
| 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: |
$\frac{46\!\cdots\!67}{24\!\cdots\!65}a^{17}-\frac{30\!\cdots\!17}{24\!\cdots\!65}a^{16}+\frac{19\!\cdots\!78}{49\!\cdots\!13}a^{15}-\frac{34\!\cdots\!76}{49\!\cdots\!13}a^{14}+\frac{54\!\cdots\!03}{49\!\cdots\!13}a^{13}-\frac{58\!\cdots\!92}{24\!\cdots\!65}a^{12}+\frac{13\!\cdots\!99}{24\!\cdots\!65}a^{11}-\frac{15\!\cdots\!28}{24\!\cdots\!65}a^{10}+\frac{11\!\cdots\!02}{24\!\cdots\!65}a^{9}+\frac{13\!\cdots\!89}{49\!\cdots\!13}a^{8}-\frac{13\!\cdots\!72}{24\!\cdots\!65}a^{7}-\frac{90\!\cdots\!78}{24\!\cdots\!65}a^{6}+\frac{26\!\cdots\!86}{24\!\cdots\!65}a^{5}-\frac{74\!\cdots\!77}{24\!\cdots\!65}a^{4}+\frac{16\!\cdots\!24}{49\!\cdots\!13}a^{3}-\frac{27\!\cdots\!26}{49\!\cdots\!13}a^{2}+\frac{70\!\cdots\!76}{24\!\cdots\!65}a-\frac{29\!\cdots\!05}{49\!\cdots\!13}$, $\frac{58\!\cdots\!67}{49\!\cdots\!30}a^{17}-\frac{42\!\cdots\!59}{49\!\cdots\!30}a^{16}+\frac{70\!\cdots\!18}{24\!\cdots\!65}a^{15}-\frac{13\!\cdots\!56}{24\!\cdots\!65}a^{14}+\frac{20\!\cdots\!48}{24\!\cdots\!65}a^{13}-\frac{42\!\cdots\!22}{24\!\cdots\!65}a^{12}+\frac{19\!\cdots\!85}{49\!\cdots\!13}a^{11}-\frac{12\!\cdots\!98}{24\!\cdots\!65}a^{10}+\frac{17\!\cdots\!32}{49\!\cdots\!13}a^{9}+\frac{39\!\cdots\!03}{24\!\cdots\!65}a^{8}-\frac{26\!\cdots\!56}{49\!\cdots\!13}a^{7}-\frac{54\!\cdots\!93}{24\!\cdots\!65}a^{6}+\frac{48\!\cdots\!83}{49\!\cdots\!30}a^{5}-\frac{15\!\cdots\!59}{49\!\cdots\!30}a^{4}-\frac{11\!\cdots\!31}{24\!\cdots\!65}a^{3}-\frac{12\!\cdots\!73}{49\!\cdots\!30}a^{2}-\frac{52\!\cdots\!17}{24\!\cdots\!65}a-\frac{11\!\cdots\!31}{49\!\cdots\!30}$, $\frac{10\!\cdots\!41}{98\!\cdots\!60}a^{17}-\frac{54\!\cdots\!79}{98\!\cdots\!60}a^{16}+\frac{30\!\cdots\!44}{24\!\cdots\!65}a^{15}-\frac{49\!\cdots\!53}{49\!\cdots\!30}a^{14}+\frac{71\!\cdots\!01}{49\!\cdots\!30}a^{13}-\frac{15\!\cdots\!38}{24\!\cdots\!65}a^{12}+\frac{63\!\cdots\!09}{49\!\cdots\!30}a^{11}+\frac{16\!\cdots\!31}{49\!\cdots\!30}a^{10}-\frac{27\!\cdots\!81}{24\!\cdots\!65}a^{9}+\frac{17\!\cdots\!43}{49\!\cdots\!13}a^{8}+\frac{66\!\cdots\!51}{98\!\cdots\!26}a^{7}-\frac{30\!\cdots\!34}{49\!\cdots\!13}a^{6}+\frac{86\!\cdots\!47}{98\!\cdots\!60}a^{5}+\frac{63\!\cdots\!93}{98\!\cdots\!60}a^{4}+\frac{27\!\cdots\!27}{98\!\cdots\!26}a^{3}+\frac{27\!\cdots\!11}{19\!\cdots\!52}a^{2}+\frac{17\!\cdots\!73}{49\!\cdots\!30}a+\frac{60\!\cdots\!37}{98\!\cdots\!60}$, $\frac{16\!\cdots\!43}{98\!\cdots\!60}a^{17}-\frac{10\!\cdots\!69}{98\!\cdots\!60}a^{16}+\frac{77\!\cdots\!41}{24\!\cdots\!65}a^{15}-\frac{27\!\cdots\!13}{49\!\cdots\!30}a^{14}+\frac{43\!\cdots\!19}{49\!\cdots\!30}a^{13}-\frac{48\!\cdots\!38}{24\!\cdots\!65}a^{12}+\frac{20\!\cdots\!37}{49\!\cdots\!30}a^{11}-\frac{45\!\cdots\!71}{98\!\cdots\!26}a^{10}+\frac{90\!\cdots\!44}{24\!\cdots\!65}a^{9}+\frac{60\!\cdots\!99}{24\!\cdots\!65}a^{8}-\frac{17\!\cdots\!61}{49\!\cdots\!30}a^{7}-\frac{82\!\cdots\!12}{24\!\cdots\!65}a^{6}+\frac{66\!\cdots\!13}{98\!\cdots\!60}a^{5}-\frac{16\!\cdots\!13}{98\!\cdots\!60}a^{4}+\frac{24\!\cdots\!07}{49\!\cdots\!30}a^{3}+\frac{89\!\cdots\!61}{98\!\cdots\!60}a^{2}+\frac{32\!\cdots\!13}{49\!\cdots\!30}a+\frac{76\!\cdots\!11}{98\!\cdots\!60}$, $\frac{30\!\cdots\!77}{49\!\cdots\!30}a^{17}-\frac{13\!\cdots\!71}{24\!\cdots\!65}a^{16}+\frac{50\!\cdots\!91}{24\!\cdots\!65}a^{15}-\frac{11\!\cdots\!16}{24\!\cdots\!65}a^{14}+\frac{35\!\cdots\!91}{49\!\cdots\!13}a^{13}-\frac{32\!\cdots\!09}{24\!\cdots\!65}a^{12}+\frac{73\!\cdots\!08}{24\!\cdots\!65}a^{11}-\frac{23\!\cdots\!68}{49\!\cdots\!13}a^{10}+\frac{10\!\cdots\!48}{24\!\cdots\!65}a^{9}-\frac{88\!\cdots\!54}{24\!\cdots\!65}a^{8}-\frac{12\!\cdots\!78}{24\!\cdots\!65}a^{7}+\frac{91\!\cdots\!23}{49\!\cdots\!13}a^{6}+\frac{36\!\cdots\!29}{49\!\cdots\!30}a^{5}-\frac{17\!\cdots\!46}{24\!\cdots\!65}a^{4}+\frac{81\!\cdots\!06}{24\!\cdots\!65}a^{3}-\frac{56\!\cdots\!29}{49\!\cdots\!30}a^{2}+\frac{20\!\cdots\!71}{98\!\cdots\!26}a-\frac{45\!\cdots\!09}{49\!\cdots\!13}$, $\frac{81\!\cdots\!79}{24\!\cdots\!86}a^{17}-\frac{11\!\cdots\!33}{60\!\cdots\!65}a^{16}+\frac{29\!\cdots\!47}{60\!\cdots\!65}a^{15}-\frac{37\!\cdots\!92}{60\!\cdots\!65}a^{14}+\frac{10\!\cdots\!26}{12\!\cdots\!93}a^{13}-\frac{15\!\cdots\!86}{60\!\cdots\!65}a^{12}+\frac{34\!\cdots\!81}{60\!\cdots\!65}a^{11}-\frac{16\!\cdots\!23}{60\!\cdots\!65}a^{10}-\frac{38\!\cdots\!12}{60\!\cdots\!65}a^{9}+\frac{63\!\cdots\!82}{60\!\cdots\!65}a^{8}-\frac{19\!\cdots\!72}{60\!\cdots\!65}a^{7}-\frac{18\!\cdots\!93}{12\!\cdots\!93}a^{6}+\frac{24\!\cdots\!67}{24\!\cdots\!86}a^{5}+\frac{74\!\cdots\!92}{60\!\cdots\!65}a^{4}+\frac{30\!\cdots\!26}{60\!\cdots\!65}a^{3}+\frac{31\!\cdots\!27}{12\!\cdots\!30}a^{2}+\frac{11\!\cdots\!53}{12\!\cdots\!30}a+\frac{49\!\cdots\!62}{60\!\cdots\!65}$, $\frac{93\!\cdots\!09}{48\!\cdots\!72}a^{17}-\frac{60\!\cdots\!85}{48\!\cdots\!72}a^{16}+\frac{23\!\cdots\!09}{60\!\cdots\!65}a^{15}-\frac{78\!\cdots\!01}{12\!\cdots\!30}a^{14}+\frac{11\!\cdots\!93}{12\!\cdots\!30}a^{13}-\frac{12\!\cdots\!29}{60\!\cdots\!65}a^{12}+\frac{58\!\cdots\!73}{12\!\cdots\!30}a^{11}-\frac{62\!\cdots\!03}{12\!\cdots\!30}a^{10}+\frac{14\!\cdots\!89}{60\!\cdots\!65}a^{9}+\frac{31\!\cdots\!22}{60\!\cdots\!65}a^{8}-\frac{82\!\cdots\!67}{12\!\cdots\!30}a^{7}-\frac{34\!\cdots\!62}{60\!\cdots\!65}a^{6}+\frac{30\!\cdots\!63}{24\!\cdots\!60}a^{5}-\frac{16\!\cdots\!01}{24\!\cdots\!60}a^{4}-\frac{12\!\cdots\!21}{12\!\cdots\!30}a^{3}+\frac{57\!\cdots\!27}{24\!\cdots\!60}a^{2}-\frac{41\!\cdots\!66}{60\!\cdots\!65}a+\frac{79\!\cdots\!79}{24\!\cdots\!60}$, $\frac{17\!\cdots\!99}{98\!\cdots\!60}a^{17}-\frac{11\!\cdots\!07}{98\!\cdots\!60}a^{16}+\frac{17\!\cdots\!29}{49\!\cdots\!13}a^{15}-\frac{28\!\cdots\!53}{49\!\cdots\!30}a^{14}+\frac{43\!\cdots\!51}{49\!\cdots\!30}a^{13}-\frac{49\!\cdots\!38}{24\!\cdots\!65}a^{12}+\frac{22\!\cdots\!57}{49\!\cdots\!30}a^{11}-\frac{23\!\cdots\!53}{49\!\cdots\!30}a^{10}+\frac{13\!\cdots\!18}{49\!\cdots\!13}a^{9}+\frac{99\!\cdots\!68}{24\!\cdots\!65}a^{8}-\frac{25\!\cdots\!49}{49\!\cdots\!30}a^{7}-\frac{23\!\cdots\!15}{49\!\cdots\!13}a^{6}+\frac{98\!\cdots\!33}{98\!\cdots\!60}a^{5}-\frac{72\!\cdots\!91}{98\!\cdots\!60}a^{4}+\frac{14\!\cdots\!35}{98\!\cdots\!26}a^{3}+\frac{59\!\cdots\!09}{19\!\cdots\!52}a^{2}+\frac{78\!\cdots\!47}{24\!\cdots\!65}a+\frac{75\!\cdots\!13}{98\!\cdots\!60}$
| 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: | \( 23621.7133529 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 23621.7133529 \cdot 2}{14\cdot\sqrt{105226667788517176205647}}\cr\approx \mathstrut & 0.158770298535 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 17*x^16 - 26*x^15 + 40*x^14 - 98*x^13 + 214*x^12 - 180*x^11 + 94*x^10 + 240*x^9 - 166*x^8 - 346*x^7 + 387*x^6 + 144*x^5 + 183*x^4 + 45*x^3 + 21*x^2 + 3*x + 1)
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 + 17*x^16 - 26*x^15 + 40*x^14 - 98*x^13 + 214*x^12 - 180*x^11 + 94*x^10 + 240*x^9 - 166*x^8 - 346*x^7 + 387*x^6 + 144*x^5 + 183*x^4 + 45*x^3 + 21*x^2 + 3*x + 1, 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 + 17*x^16 - 26*x^15 + 40*x^14 - 98*x^13 + 214*x^12 - 180*x^11 + 94*x^10 + 240*x^9 - 166*x^8 - 346*x^7 + 387*x^6 + 144*x^5 + 183*x^4 + 45*x^3 + 21*x^2 + 3*x + 1);
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 + 17*x^16 - 26*x^15 + 40*x^14 - 98*x^13 + 214*x^12 - 180*x^11 + 94*x^10 + 240*x^9 - 166*x^8 - 346*x^7 + 387*x^6 + 144*x^5 + 183*x^4 + 45*x^3 + 21*x^2 + 3*x + 1);
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
| \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), 3.3.2597.1, \(\Q(\zeta_{7})\), 6.0.47210863.1, 9.9.17515230173.1 |
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: | 12.0.127773131200820329.1 |
| Degree 18 sibling: | deg 18 |
| Minimal sibling: | 12.0.127773131200820329.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}{,}\,{\href{/padicField/2.3.0.1}{3} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{3}$ | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.18.15.5 | $x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98757 x^{12} + 161784 x^{11} + 246024 x^{10} + 264920 x^{9} + 530640 x^{8} + 156384 x^{7} - 1885725 x^{6} - 6133212 x^{5} - 3645540 x^{4} + 5968464 x^{3} + 5011344 x^{2} + 1820448 x + 2358791$ | $6$ | $3$ | $15$ | $C_6 \times C_3$ | $[\ ]_{6}^{3}$ |
|
\(53\)
| 53.3.0.1 | $x^{3} + 3 x + 51$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 53.3.0.1 | $x^{3} + 3 x + 51$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 53.6.3.1 | $x^{6} + 7314 x^{5} + 17831697 x^{4} + 14491896314 x^{3} + 998947242 x^{2} + 44403234204 x + 739971484841$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 53.6.3.1 | $x^{6} + 7314 x^{5} + 17831697 x^{4} + 14491896314 x^{3} + 998947242 x^{2} + 44403234204 x + 739971484841$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |