Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 23*x^16 + 300*x^15 + 515*x^14 - 4561*x^13 - 10821*x^12 + 31931*x^11 + 128094*x^10 - 8849*x^9 - 656070*x^8 - 1052332*x^7 + 297296*x^6 + 3208702*x^5 + 5247365*x^4 + 4745832*x^3 + 2696725*x^2 + 963650*x + 201125)
gp: K = bnfinit(y^18 - 9*y^17 - 23*y^16 + 300*y^15 + 515*y^14 - 4561*y^13 - 10821*y^12 + 31931*y^11 + 128094*y^10 - 8849*y^9 - 656070*y^8 - 1052332*y^7 + 297296*y^6 + 3208702*y^5 + 5247365*y^4 + 4745832*y^3 + 2696725*y^2 + 963650*y + 201125, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 - 23*x^16 + 300*x^15 + 515*x^14 - 4561*x^13 - 10821*x^12 + 31931*x^11 + 128094*x^10 - 8849*x^9 - 656070*x^8 - 1052332*x^7 + 297296*x^6 + 3208702*x^5 + 5247365*x^4 + 4745832*x^3 + 2696725*x^2 + 963650*x + 201125);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 9*x^17 - 23*x^16 + 300*x^15 + 515*x^14 - 4561*x^13 - 10821*x^12 + 31931*x^11 + 128094*x^10 - 8849*x^9 - 656070*x^8 - 1052332*x^7 + 297296*x^6 + 3208702*x^5 + 5247365*x^4 + 4745832*x^3 + 2696725*x^2 + 963650*x + 201125)
\( x^{18} - 9 x^{17} - 23 x^{16} + 300 x^{15} + 515 x^{14} - 4561 x^{13} - 10821 x^{12} + 31931 x^{11} + \cdots + 201125 \)
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: | $[4, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-14616700658977641856215848806693687375\)
\(\medspace = -\,5^{3}\cdot 7^{12}\cdot 41^{4}\cdot 419^{3}\cdot 449^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(116.07\) | 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: | $5^{1/2}7^{2/3}41^{2/3}419^{1/2}449^{2/3}\approx 116773.46634277423$ | ||
| Ramified primes: |
\(5\), \(7\), \(41\), \(419\), \(449\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-2095}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{105}a^{16}-\frac{4}{105}a^{15}+\frac{22}{105}a^{14}+\frac{3}{7}a^{13}-\frac{1}{3}a^{12}-\frac{26}{105}a^{11}+\frac{4}{105}a^{10}+\frac{1}{5}a^{9}+\frac{44}{105}a^{8}-\frac{19}{105}a^{7}+\frac{1}{21}a^{6}+\frac{4}{15}a^{5}-\frac{19}{105}a^{4}+\frac{52}{105}a^{3}-\frac{8}{21}a^{2}-\frac{11}{35}a-\frac{5}{21}$, $\frac{1}{12\!\cdots\!75}a^{17}+\frac{52\!\cdots\!26}{12\!\cdots\!75}a^{16}-\frac{47\!\cdots\!84}{18\!\cdots\!25}a^{15}-\frac{93\!\cdots\!46}{25\!\cdots\!95}a^{14}+\frac{57\!\cdots\!28}{25\!\cdots\!95}a^{13}-\frac{10\!\cdots\!04}{14\!\cdots\!75}a^{12}-\frac{19\!\cdots\!77}{42\!\cdots\!25}a^{11}-\frac{53\!\cdots\!54}{12\!\cdots\!75}a^{10}-\frac{26\!\cdots\!71}{12\!\cdots\!75}a^{9}+\frac{35\!\cdots\!22}{42\!\cdots\!25}a^{8}+\frac{13\!\cdots\!72}{28\!\cdots\!55}a^{7}-\frac{54\!\cdots\!82}{12\!\cdots\!75}a^{6}+\frac{35\!\cdots\!76}{12\!\cdots\!75}a^{5}-\frac{57\!\cdots\!34}{18\!\cdots\!25}a^{4}+\frac{13\!\cdots\!07}{25\!\cdots\!95}a^{3}-\frac{49\!\cdots\!68}{12\!\cdots\!75}a^{2}-\frac{89\!\cdots\!58}{36\!\cdots\!85}a-\frac{14\!\cdots\!03}{50\!\cdots\!19}$
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
Trivial group, which has order $1$ (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: | $10$ | 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: |
$\frac{59\!\cdots\!24}{76\!\cdots\!25}a^{17}-\frac{65\!\cdots\!76}{76\!\cdots\!25}a^{16}-\frac{32\!\cdots\!12}{76\!\cdots\!25}a^{15}+\frac{41\!\cdots\!51}{15\!\cdots\!65}a^{14}-\frac{13\!\cdots\!23}{15\!\cdots\!65}a^{13}-\frac{11\!\cdots\!88}{25\!\cdots\!75}a^{12}-\frac{12\!\cdots\!48}{25\!\cdots\!75}a^{11}+\frac{31\!\cdots\!04}{76\!\cdots\!25}a^{10}+\frac{27\!\cdots\!21}{76\!\cdots\!25}a^{9}-\frac{17\!\cdots\!74}{85\!\cdots\!25}a^{8}-\frac{27\!\cdots\!63}{73\!\cdots\!65}a^{7}+\frac{23\!\cdots\!32}{76\!\cdots\!25}a^{6}+\frac{10\!\cdots\!74}{76\!\cdots\!25}a^{5}+\frac{67\!\cdots\!38}{76\!\cdots\!25}a^{4}-\frac{11\!\cdots\!67}{15\!\cdots\!65}a^{3}-\frac{10\!\cdots\!32}{76\!\cdots\!25}a^{2}-\frac{11\!\cdots\!69}{15\!\cdots\!65}a-\frac{56\!\cdots\!70}{30\!\cdots\!53}$, $\frac{59\!\cdots\!24}{76\!\cdots\!25}a^{17}-\frac{65\!\cdots\!76}{76\!\cdots\!25}a^{16}-\frac{32\!\cdots\!12}{76\!\cdots\!25}a^{15}+\frac{41\!\cdots\!51}{15\!\cdots\!65}a^{14}-\frac{13\!\cdots\!23}{15\!\cdots\!65}a^{13}-\frac{11\!\cdots\!88}{25\!\cdots\!75}a^{12}-\frac{12\!\cdots\!48}{25\!\cdots\!75}a^{11}+\frac{31\!\cdots\!04}{76\!\cdots\!25}a^{10}+\frac{27\!\cdots\!21}{76\!\cdots\!25}a^{9}-\frac{17\!\cdots\!74}{85\!\cdots\!25}a^{8}-\frac{27\!\cdots\!63}{73\!\cdots\!65}a^{7}+\frac{23\!\cdots\!32}{76\!\cdots\!25}a^{6}+\frac{10\!\cdots\!74}{76\!\cdots\!25}a^{5}+\frac{67\!\cdots\!38}{76\!\cdots\!25}a^{4}-\frac{11\!\cdots\!67}{15\!\cdots\!65}a^{3}-\frac{10\!\cdots\!32}{76\!\cdots\!25}a^{2}-\frac{11\!\cdots\!69}{15\!\cdots\!65}a-\frac{25\!\cdots\!17}{30\!\cdots\!53}$, $\frac{39\!\cdots\!24}{12\!\cdots\!75}a^{17}-\frac{47\!\cdots\!76}{12\!\cdots\!75}a^{16}+\frac{36\!\cdots\!88}{12\!\cdots\!75}a^{15}+\frac{23\!\cdots\!41}{25\!\cdots\!95}a^{14}-\frac{29\!\cdots\!68}{25\!\cdots\!95}a^{13}-\frac{55\!\cdots\!38}{42\!\cdots\!25}a^{12}+\frac{88\!\cdots\!84}{14\!\cdots\!75}a^{11}+\frac{14\!\cdots\!79}{12\!\cdots\!75}a^{10}+\frac{85\!\cdots\!21}{12\!\cdots\!75}a^{9}-\frac{20\!\cdots\!47}{42\!\cdots\!25}a^{8}-\frac{10\!\cdots\!48}{12\!\cdots\!95}a^{7}+\frac{39\!\cdots\!57}{12\!\cdots\!75}a^{6}+\frac{31\!\cdots\!74}{12\!\cdots\!75}a^{5}+\frac{38\!\cdots\!13}{12\!\cdots\!75}a^{4}+\frac{41\!\cdots\!13}{25\!\cdots\!95}a^{3}+\frac{24\!\cdots\!43}{12\!\cdots\!75}a^{2}-\frac{27\!\cdots\!34}{25\!\cdots\!95}a+\frac{27\!\cdots\!50}{50\!\cdots\!19}$, $\frac{19\!\cdots\!67}{14\!\cdots\!75}a^{17}-\frac{22\!\cdots\!28}{14\!\cdots\!75}a^{16}+\frac{20\!\cdots\!09}{14\!\cdots\!75}a^{15}+\frac{21\!\cdots\!33}{56\!\cdots\!91}a^{14}-\frac{12\!\cdots\!14}{28\!\cdots\!55}a^{13}-\frac{70\!\cdots\!87}{14\!\cdots\!75}a^{12}+\frac{17\!\cdots\!68}{14\!\cdots\!75}a^{11}+\frac{57\!\cdots\!02}{14\!\cdots\!75}a^{10}+\frac{60\!\cdots\!48}{14\!\cdots\!75}a^{9}-\frac{19\!\cdots\!08}{14\!\cdots\!75}a^{8}-\frac{16\!\cdots\!44}{40\!\cdots\!65}a^{7}-\frac{28\!\cdots\!19}{14\!\cdots\!75}a^{6}+\frac{11\!\cdots\!32}{14\!\cdots\!75}a^{5}+\frac{25\!\cdots\!09}{14\!\cdots\!75}a^{4}+\frac{53\!\cdots\!91}{28\!\cdots\!55}a^{3}+\frac{16\!\cdots\!69}{14\!\cdots\!75}a^{2}+\frac{25\!\cdots\!69}{56\!\cdots\!91}a+\frac{48\!\cdots\!37}{56\!\cdots\!91}$, $\frac{26\!\cdots\!99}{25\!\cdots\!95}a^{17}-\frac{52\!\cdots\!94}{50\!\cdots\!19}a^{16}-\frac{42\!\cdots\!06}{25\!\cdots\!95}a^{15}+\frac{86\!\cdots\!27}{25\!\cdots\!95}a^{14}+\frac{13\!\cdots\!40}{50\!\cdots\!19}a^{13}-\frac{45\!\cdots\!73}{84\!\cdots\!65}a^{12}-\frac{40\!\cdots\!30}{56\!\cdots\!91}a^{11}+\frac{11\!\cdots\!93}{25\!\cdots\!95}a^{10}+\frac{26\!\cdots\!57}{25\!\cdots\!95}a^{9}-\frac{16\!\cdots\!12}{12\!\cdots\!95}a^{8}-\frac{55\!\cdots\!58}{84\!\cdots\!65}a^{7}-\frac{10\!\cdots\!33}{25\!\cdots\!95}a^{6}+\frac{30\!\cdots\!07}{25\!\cdots\!95}a^{5}+\frac{65\!\cdots\!09}{25\!\cdots\!95}a^{4}+\frac{57\!\cdots\!92}{25\!\cdots\!95}a^{3}+\frac{41\!\cdots\!59}{36\!\cdots\!85}a^{2}+\frac{92\!\cdots\!57}{25\!\cdots\!95}a+\frac{16\!\cdots\!92}{50\!\cdots\!19}$, $\frac{15\!\cdots\!46}{12\!\cdots\!75}a^{17}-\frac{15\!\cdots\!39}{12\!\cdots\!75}a^{16}-\frac{21\!\cdots\!83}{12\!\cdots\!75}a^{15}+\frac{19\!\cdots\!41}{50\!\cdots\!19}a^{14}+\frac{90\!\cdots\!59}{36\!\cdots\!85}a^{13}-\frac{85\!\cdots\!09}{14\!\cdots\!75}a^{12}-\frac{30\!\cdots\!47}{42\!\cdots\!25}a^{11}+\frac{92\!\cdots\!93}{18\!\cdots\!25}a^{10}+\frac{14\!\cdots\!49}{12\!\cdots\!75}a^{9}-\frac{70\!\cdots\!68}{42\!\cdots\!25}a^{8}-\frac{21\!\cdots\!21}{28\!\cdots\!55}a^{7}-\frac{77\!\cdots\!21}{18\!\cdots\!25}a^{6}+\frac{19\!\cdots\!16}{12\!\cdots\!75}a^{5}+\frac{40\!\cdots\!17}{12\!\cdots\!75}a^{4}+\frac{55\!\cdots\!28}{25\!\cdots\!95}a^{3}-\frac{20\!\cdots\!28}{12\!\cdots\!75}a^{2}-\frac{55\!\cdots\!52}{50\!\cdots\!19}a-\frac{37\!\cdots\!32}{72\!\cdots\!17}$, $\frac{53\!\cdots\!62}{60\!\cdots\!75}a^{17}-\frac{44\!\cdots\!91}{42\!\cdots\!25}a^{16}+\frac{39\!\cdots\!83}{42\!\cdots\!25}a^{15}+\frac{70\!\cdots\!97}{28\!\cdots\!55}a^{14}-\frac{24\!\cdots\!48}{84\!\cdots\!65}a^{13}-\frac{20\!\cdots\!82}{60\!\cdots\!75}a^{12}+\frac{41\!\cdots\!71}{42\!\cdots\!25}a^{11}+\frac{40\!\cdots\!63}{14\!\cdots\!75}a^{10}+\frac{15\!\cdots\!48}{60\!\cdots\!75}a^{9}-\frac{47\!\cdots\!31}{42\!\cdots\!25}a^{8}-\frac{22\!\cdots\!63}{84\!\cdots\!65}a^{7}-\frac{18\!\cdots\!38}{42\!\cdots\!25}a^{6}+\frac{34\!\cdots\!12}{60\!\cdots\!75}a^{5}+\frac{44\!\cdots\!33}{42\!\cdots\!25}a^{4}+\frac{84\!\cdots\!28}{84\!\cdots\!65}a^{3}+\frac{80\!\cdots\!71}{14\!\cdots\!75}a^{2}+\frac{17\!\cdots\!16}{84\!\cdots\!65}a+\frac{23\!\cdots\!26}{56\!\cdots\!91}$, $\frac{27\!\cdots\!29}{12\!\cdots\!75}a^{17}-\frac{34\!\cdots\!46}{12\!\cdots\!75}a^{16}+\frac{38\!\cdots\!48}{12\!\cdots\!75}a^{15}+\frac{16\!\cdots\!71}{25\!\cdots\!95}a^{14}-\frac{31\!\cdots\!18}{25\!\cdots\!95}a^{13}-\frac{37\!\cdots\!73}{42\!\cdots\!25}a^{12}+\frac{48\!\cdots\!42}{42\!\cdots\!25}a^{11}+\frac{95\!\cdots\!09}{12\!\cdots\!75}a^{10}-\frac{27\!\cdots\!34}{12\!\cdots\!75}a^{9}-\frac{51\!\cdots\!79}{14\!\cdots\!75}a^{8}-\frac{32\!\cdots\!73}{12\!\cdots\!95}a^{7}+\frac{84\!\cdots\!72}{12\!\cdots\!75}a^{6}+\frac{16\!\cdots\!54}{12\!\cdots\!75}a^{5}+\frac{88\!\cdots\!23}{12\!\cdots\!75}a^{4}-\frac{72\!\cdots\!57}{25\!\cdots\!95}a^{3}-\frac{80\!\cdots\!22}{12\!\cdots\!75}a^{2}-\frac{87\!\cdots\!49}{25\!\cdots\!95}a-\frac{67\!\cdots\!97}{50\!\cdots\!19}$, $\frac{19\!\cdots\!57}{14\!\cdots\!75}a^{17}-\frac{72\!\cdots\!03}{14\!\cdots\!75}a^{16}-\frac{78\!\cdots\!01}{14\!\cdots\!75}a^{15}+\frac{29\!\cdots\!44}{28\!\cdots\!55}a^{14}+\frac{33\!\cdots\!41}{28\!\cdots\!55}a^{13}+\frac{30\!\cdots\!73}{14\!\cdots\!75}a^{12}-\frac{24\!\cdots\!01}{20\!\cdots\!25}a^{11}-\frac{31\!\cdots\!93}{14\!\cdots\!75}a^{10}+\frac{56\!\cdots\!93}{14\!\cdots\!75}a^{9}+\frac{26\!\cdots\!47}{14\!\cdots\!75}a^{8}+\frac{53\!\cdots\!89}{28\!\cdots\!55}a^{7}-\frac{37\!\cdots\!99}{14\!\cdots\!75}a^{6}-\frac{14\!\cdots\!73}{14\!\cdots\!75}a^{5}-\frac{20\!\cdots\!01}{14\!\cdots\!75}a^{4}-\frac{67\!\cdots\!00}{56\!\cdots\!91}a^{3}-\frac{91\!\cdots\!51}{14\!\cdots\!75}a^{2}-\frac{62\!\cdots\!01}{28\!\cdots\!55}a-\frac{25\!\cdots\!81}{56\!\cdots\!91}$, $\frac{27\!\cdots\!08}{20\!\cdots\!25}a^{17}-\frac{50\!\cdots\!87}{42\!\cdots\!25}a^{16}-\frac{14\!\cdots\!39}{42\!\cdots\!25}a^{15}+\frac{71\!\cdots\!68}{16\!\cdots\!73}a^{14}+\frac{22\!\cdots\!73}{28\!\cdots\!55}a^{13}-\frac{39\!\cdots\!39}{60\!\cdots\!75}a^{12}-\frac{68\!\cdots\!03}{42\!\cdots\!25}a^{11}+\frac{20\!\cdots\!83}{42\!\cdots\!25}a^{10}+\frac{38\!\cdots\!27}{20\!\cdots\!25}a^{9}-\frac{19\!\cdots\!82}{42\!\cdots\!25}a^{8}-\frac{87\!\cdots\!22}{84\!\cdots\!65}a^{7}-\frac{58\!\cdots\!01}{42\!\cdots\!25}a^{6}+\frac{63\!\cdots\!79}{60\!\cdots\!75}a^{5}+\frac{21\!\cdots\!86}{42\!\cdots\!25}a^{4}+\frac{53\!\cdots\!34}{84\!\cdots\!65}a^{3}+\frac{17\!\cdots\!26}{42\!\cdots\!25}a^{2}+\frac{96\!\cdots\!74}{56\!\cdots\!91}a+\frac{67\!\cdots\!83}{16\!\cdots\!73}$
| 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: | \( 625949725971 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{7}\cdot 625949725971 \cdot 1}{2\cdot\sqrt{14616700658977641856215848806693687375}}\cr\approx \mathstrut & 0.506365190150983 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 23*x^16 + 300*x^15 + 515*x^14 - 4561*x^13 - 10821*x^12 + 31931*x^11 + 128094*x^10 - 8849*x^9 - 656070*x^8 - 1052332*x^7 + 297296*x^6 + 3208702*x^5 + 5247365*x^4 + 4745832*x^3 + 2696725*x^2 + 963650*x + 201125)
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 - 9*x^17 - 23*x^16 + 300*x^15 + 515*x^14 - 4561*x^13 - 10821*x^12 + 31931*x^11 + 128094*x^10 - 8849*x^9 - 656070*x^8 - 1052332*x^7 + 297296*x^6 + 3208702*x^5 + 5247365*x^4 + 4745832*x^3 + 2696725*x^2 + 963650*x + 201125, 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 - 9*x^17 - 23*x^16 + 300*x^15 + 515*x^14 - 4561*x^13 - 10821*x^12 + 31931*x^11 + 128094*x^10 - 8849*x^9 - 656070*x^8 - 1052332*x^7 + 297296*x^6 + 3208702*x^5 + 5247365*x^4 + 4745832*x^3 + 2696725*x^2 + 963650*x + 201125);
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 - 9*x^17 - 23*x^16 + 300*x^15 + 515*x^14 - 4561*x^13 - 10821*x^12 + 31931*x^11 + 128094*x^10 - 8849*x^9 - 656070*x^8 - 1052332*x^7 + 297296*x^6 + 3208702*x^5 + 5247365*x^4 + 4745832*x^3 + 2696725*x^2 + 963650*x + 201125);
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
$C_3^6.C_2\wr C_6$ (as 18T858):
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 279936 |
| The 159 conjugacy class representatives for $C_3^6.C_2\wr C_6$ |
| Character table for $C_3^6.C_2\wr C_6$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 6.4.1006019.1 |
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 18 sibling: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 18.8.3838671405368745874099609375.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{4}$ | R | R | $18$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }$ | $18$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{8}$ | $18$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.9.0.1 | $x^{9} + 2 x^{3} + x + 3$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
|
\(7\)
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
|
\(41\)
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.3.0.1 | $x^{3} + x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 41.3.0.1 | $x^{3} + x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 41.6.4.2 | $x^{6} - 1804 x^{3} - 4557191$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
|
\(419\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $2$ | $3$ | $3$ | ||||
|
\(449\)
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $3$ | $2$ | $4$ |