Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 3*x^16 + 36*x^15 - 72*x^14 + 24*x^13 + 677*x^12 - 2154*x^11 + 650*x^10 - 12520*x^9 + 93111*x^8 - 184794*x^7 + 51751*x^6 + 253950*x^5 - 276628*x^4 - 32494*x^3 + 217588*x^2 - 143080*x + 34300)
gp: K = bnfinit(y^18 - 6*y^17 + 3*y^16 + 36*y^15 - 72*y^14 + 24*y^13 + 677*y^12 - 2154*y^11 + 650*y^10 - 12520*y^9 + 93111*y^8 - 184794*y^7 + 51751*y^6 + 253950*y^5 - 276628*y^4 - 32494*y^3 + 217588*y^2 - 143080*y + 34300, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 3*x^16 + 36*x^15 - 72*x^14 + 24*x^13 + 677*x^12 - 2154*x^11 + 650*x^10 - 12520*x^9 + 93111*x^8 - 184794*x^7 + 51751*x^6 + 253950*x^5 - 276628*x^4 - 32494*x^3 + 217588*x^2 - 143080*x + 34300);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 + 3*x^16 + 36*x^15 - 72*x^14 + 24*x^13 + 677*x^12 - 2154*x^11 + 650*x^10 - 12520*x^9 + 93111*x^8 - 184794*x^7 + 51751*x^6 + 253950*x^5 - 276628*x^4 - 32494*x^3 + 217588*x^2 - 143080*x + 34300)
\( x^{18} - 6 x^{17} + 3 x^{16} + 36 x^{15} - 72 x^{14} + 24 x^{13} + 677 x^{12} - 2154 x^{11} + \cdots + 34300 \)
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: |
\(-21423624263414745793258586112\)
\(\medspace = -\,2^{26}\cdot 3^{9}\cdot 7^{6}\cdot 13^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(37.49\) | 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: | $2^{11/6}3^{1/2}7^{1/2}13^{5/6}\approx 138.44717539350094$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{35}a^{13}-\frac{3}{35}a^{12}-\frac{9}{35}a^{11}-\frac{9}{35}a^{10}+\frac{16}{35}a^{9}+\frac{6}{35}a^{8}-\frac{13}{35}a^{7}-\frac{11}{35}a^{6}+\frac{11}{35}a^{5}+\frac{2}{5}a^{4}+\frac{16}{35}a^{3}-\frac{12}{35}a^{2}-\frac{2}{5}a$, $\frac{1}{35}a^{14}+\frac{17}{35}a^{12}-\frac{1}{35}a^{11}-\frac{11}{35}a^{10}-\frac{16}{35}a^{9}+\frac{1}{7}a^{8}-\frac{3}{7}a^{7}+\frac{13}{35}a^{6}+\frac{12}{35}a^{5}-\frac{12}{35}a^{4}+\frac{1}{35}a^{3}-\frac{3}{7}a^{2}-\frac{1}{5}a$, $\frac{1}{70}a^{15}-\frac{1}{70}a^{13}+\frac{9}{35}a^{12}-\frac{12}{35}a^{11}+\frac{3}{35}a^{10}-\frac{3}{70}a^{9}-\frac{9}{35}a^{8}+\frac{1}{35}a^{7}-\frac{1}{2}a^{5}-\frac{3}{35}a^{4}-\frac{23}{70}a^{3}+\frac{17}{35}a^{2}-\frac{2}{5}a$, $\frac{1}{490}a^{16}+\frac{1}{245}a^{15}+\frac{1}{98}a^{14}+\frac{3}{245}a^{13}+\frac{37}{245}a^{12}-\frac{1}{5}a^{11}+\frac{33}{490}a^{10}+\frac{108}{245}a^{8}-\frac{48}{245}a^{7}+\frac{153}{490}a^{6}-\frac{92}{245}a^{5}-\frac{107}{490}a^{4}-\frac{48}{245}a^{3}+\frac{1}{7}a^{2}-\frac{2}{35}a$, $\frac{1}{51\!\cdots\!90}a^{17}-\frac{50\!\cdots\!17}{51\!\cdots\!90}a^{16}-\frac{36\!\cdots\!57}{51\!\cdots\!90}a^{15}-\frac{14\!\cdots\!27}{10\!\cdots\!78}a^{14}-\frac{37\!\cdots\!26}{25\!\cdots\!45}a^{13}-\frac{82\!\cdots\!21}{51\!\cdots\!89}a^{12}-\frac{14\!\cdots\!99}{51\!\cdots\!90}a^{11}+\frac{13\!\cdots\!39}{51\!\cdots\!90}a^{10}-\frac{62\!\cdots\!26}{25\!\cdots\!45}a^{9}+\frac{16\!\cdots\!34}{37\!\cdots\!35}a^{8}+\frac{97\!\cdots\!07}{51\!\cdots\!90}a^{7}+\frac{31\!\cdots\!79}{98\!\cdots\!30}a^{6}-\frac{99\!\cdots\!99}{51\!\cdots\!90}a^{5}+\frac{30\!\cdots\!29}{51\!\cdots\!90}a^{4}+\frac{81\!\cdots\!32}{25\!\cdots\!45}a^{3}+\frac{13\!\cdots\!57}{37\!\cdots\!35}a^{2}-\frac{28\!\cdots\!31}{74\!\cdots\!27}a+\frac{18\!\cdots\!60}{10\!\cdots\!61}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{496420080872530195347761}{134848137328992242090019970} a^{17} + \frac{1157413271139649587517433}{67424068664496121045009985} a^{16} + \frac{818540017525873407977846}{67424068664496121045009985} a^{15} - \frac{7898152328111331121651218}{67424068664496121045009985} a^{14} + \frac{14401135879399958821435597}{134848137328992242090019970} a^{13} + \frac{4055843878580149773105588}{67424068664496121045009985} a^{12} - \frac{325369432107158936268641837}{134848137328992242090019970} a^{11} + \frac{316808509812809483286727682}{67424068664496121045009985} a^{10} + \frac{108907746652542353083014165}{26969627465798448418003994} a^{9} + \frac{692391939408725994949483277}{13484813732899224209001997} a^{8} - \frac{37019154862225585029674522001}{134848137328992242090019970} a^{7} + \frac{393863627550981434291005564}{1272152238952757000849245} a^{6} + \frac{15976867477599680789995031567}{67424068664496121045009985} a^{5} - \frac{8410412741412790607063356313}{13484813732899224209001997} a^{4} + \frac{22023715586511625996090874653}{134848137328992242090019970} a^{3} + \frac{24135168416560644590280382662}{67424068664496121045009985} a^{2} - \frac{21236073514182469164529569514}{67424068664496121045009985} a + \frac{1216715851185466286411991180}{13484813732899224209001997} \)
(order $6$)
| 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: |
$\frac{28\!\cdots\!33}{51\!\cdots\!90}a^{17}-\frac{13\!\cdots\!57}{51\!\cdots\!90}a^{16}-\frac{66\!\cdots\!62}{37\!\cdots\!35}a^{15}+\frac{91\!\cdots\!01}{51\!\cdots\!90}a^{14}-\frac{84\!\cdots\!09}{51\!\cdots\!90}a^{13}-\frac{23\!\cdots\!58}{25\!\cdots\!45}a^{12}+\frac{18\!\cdots\!03}{51\!\cdots\!90}a^{11}-\frac{36\!\cdots\!89}{51\!\cdots\!90}a^{10}-\frac{31\!\cdots\!89}{51\!\cdots\!90}a^{9}-\frac{39\!\cdots\!05}{51\!\cdots\!89}a^{8}+\frac{21\!\cdots\!23}{51\!\cdots\!90}a^{7}-\frac{45\!\cdots\!19}{98\!\cdots\!30}a^{6}-\frac{92\!\cdots\!99}{25\!\cdots\!45}a^{5}+\frac{69\!\cdots\!87}{74\!\cdots\!70}a^{4}-\frac{13\!\cdots\!23}{51\!\cdots\!90}a^{3}-\frac{19\!\cdots\!93}{37\!\cdots\!35}a^{2}+\frac{17\!\cdots\!46}{37\!\cdots\!35}a-\frac{14\!\cdots\!24}{10\!\cdots\!61}$, $\frac{18\!\cdots\!73}{51\!\cdots\!89}a^{17}-\frac{60\!\cdots\!39}{25\!\cdots\!45}a^{16}+\frac{92\!\cdots\!37}{74\!\cdots\!70}a^{15}+\frac{42\!\cdots\!93}{25\!\cdots\!45}a^{14}-\frac{26\!\cdots\!51}{10\!\cdots\!78}a^{13}-\frac{14\!\cdots\!82}{25\!\cdots\!45}a^{12}+\frac{62\!\cdots\!88}{25\!\cdots\!45}a^{11}-\frac{22\!\cdots\!96}{25\!\cdots\!45}a^{10}-\frac{18\!\cdots\!23}{51\!\cdots\!90}a^{9}-\frac{18\!\cdots\!92}{51\!\cdots\!89}a^{8}+\frac{96\!\cdots\!67}{25\!\cdots\!45}a^{7}-\frac{33\!\cdots\!39}{49\!\cdots\!65}a^{6}-\frac{63\!\cdots\!83}{51\!\cdots\!90}a^{5}+\frac{44\!\cdots\!04}{37\!\cdots\!35}a^{4}-\frac{29\!\cdots\!83}{51\!\cdots\!90}a^{3}-\frac{22\!\cdots\!79}{37\!\cdots\!35}a^{2}+\frac{47\!\cdots\!23}{74\!\cdots\!27}a-\frac{22\!\cdots\!57}{10\!\cdots\!61}$, $\frac{51\!\cdots\!49}{51\!\cdots\!90}a^{17}-\frac{12\!\cdots\!02}{25\!\cdots\!45}a^{16}-\frac{78\!\cdots\!36}{37\!\cdots\!35}a^{15}+\frac{17\!\cdots\!59}{51\!\cdots\!89}a^{14}-\frac{19\!\cdots\!33}{51\!\cdots\!90}a^{13}-\frac{34\!\cdots\!26}{25\!\cdots\!45}a^{12}+\frac{68\!\cdots\!89}{10\!\cdots\!78}a^{11}-\frac{37\!\cdots\!62}{25\!\cdots\!45}a^{10}-\frac{42\!\cdots\!39}{51\!\cdots\!90}a^{9}-\frac{34\!\cdots\!19}{25\!\cdots\!45}a^{8}+\frac{81\!\cdots\!95}{10\!\cdots\!78}a^{7}-\frac{50\!\cdots\!87}{49\!\cdots\!65}a^{6}-\frac{26\!\cdots\!47}{51\!\cdots\!89}a^{5}+\frac{14\!\cdots\!34}{74\!\cdots\!27}a^{4}-\frac{40\!\cdots\!79}{51\!\cdots\!90}a^{3}-\frac{41\!\cdots\!62}{37\!\cdots\!35}a^{2}+\frac{41\!\cdots\!18}{37\!\cdots\!35}a-\frac{31\!\cdots\!64}{10\!\cdots\!61}$, $\frac{17\!\cdots\!03}{10\!\cdots\!78}a^{17}-\frac{23\!\cdots\!24}{25\!\cdots\!45}a^{16}-\frac{70\!\cdots\!88}{25\!\cdots\!45}a^{15}+\frac{31\!\cdots\!96}{51\!\cdots\!89}a^{14}-\frac{51\!\cdots\!09}{74\!\cdots\!70}a^{13}-\frac{70\!\cdots\!96}{25\!\cdots\!45}a^{12}+\frac{59\!\cdots\!19}{51\!\cdots\!90}a^{11}-\frac{71\!\cdots\!92}{25\!\cdots\!45}a^{10}-\frac{15\!\cdots\!89}{10\!\cdots\!78}a^{9}-\frac{58\!\cdots\!24}{25\!\cdots\!45}a^{8}+\frac{10\!\cdots\!89}{74\!\cdots\!70}a^{7}-\frac{94\!\cdots\!74}{49\!\cdots\!65}a^{6}-\frac{25\!\cdots\!09}{25\!\cdots\!45}a^{5}+\frac{95\!\cdots\!93}{25\!\cdots\!45}a^{4}-\frac{69\!\cdots\!47}{51\!\cdots\!90}a^{3}-\frac{14\!\cdots\!02}{74\!\cdots\!27}a^{2}+\frac{72\!\cdots\!56}{37\!\cdots\!35}a-\frac{63\!\cdots\!88}{10\!\cdots\!61}$, $\frac{37\!\cdots\!66}{37\!\cdots\!35}a^{17}-\frac{12\!\cdots\!88}{25\!\cdots\!45}a^{16}-\frac{26\!\cdots\!73}{10\!\cdots\!78}a^{15}+\frac{87\!\cdots\!56}{25\!\cdots\!45}a^{14}-\frac{17\!\cdots\!89}{51\!\cdots\!90}a^{13}-\frac{42\!\cdots\!57}{25\!\cdots\!45}a^{12}+\frac{24\!\cdots\!69}{37\!\cdots\!35}a^{11}-\frac{74\!\cdots\!14}{51\!\cdots\!89}a^{10}-\frac{74\!\cdots\!13}{74\!\cdots\!70}a^{9}-\frac{70\!\cdots\!52}{51\!\cdots\!89}a^{8}+\frac{20\!\cdots\!73}{25\!\cdots\!45}a^{7}-\frac{47\!\cdots\!26}{49\!\cdots\!65}a^{6}-\frac{32\!\cdots\!19}{51\!\cdots\!90}a^{5}+\frac{49\!\cdots\!44}{25\!\cdots\!45}a^{4}-\frac{61\!\cdots\!13}{10\!\cdots\!78}a^{3}-\frac{39\!\cdots\!86}{37\!\cdots\!35}a^{2}+\frac{36\!\cdots\!81}{37\!\cdots\!35}a-\frac{30\!\cdots\!10}{10\!\cdots\!61}$, $\frac{41\!\cdots\!04}{51\!\cdots\!89}a^{17}-\frac{18\!\cdots\!13}{51\!\cdots\!90}a^{16}-\frac{16\!\cdots\!27}{51\!\cdots\!90}a^{15}+\frac{12\!\cdots\!69}{51\!\cdots\!90}a^{14}-\frac{10\!\cdots\!57}{51\!\cdots\!90}a^{13}-\frac{34\!\cdots\!31}{25\!\cdots\!45}a^{12}+\frac{13\!\cdots\!22}{25\!\cdots\!45}a^{11}-\frac{47\!\cdots\!79}{51\!\cdots\!90}a^{10}-\frac{49\!\cdots\!21}{51\!\cdots\!90}a^{9}-\frac{59\!\cdots\!66}{51\!\cdots\!89}a^{8}+\frac{14\!\cdots\!48}{25\!\cdots\!45}a^{7}-\frac{57\!\cdots\!09}{98\!\cdots\!30}a^{6}-\frac{79\!\cdots\!01}{14\!\cdots\!54}a^{5}+\frac{63\!\cdots\!09}{51\!\cdots\!90}a^{4}-\frac{27\!\cdots\!77}{10\!\cdots\!78}a^{3}-\frac{26\!\cdots\!31}{37\!\cdots\!35}a^{2}+\frac{22\!\cdots\!96}{37\!\cdots\!35}a-\frac{17\!\cdots\!29}{10\!\cdots\!61}$, $\frac{26\!\cdots\!01}{10\!\cdots\!78}a^{17}-\frac{44\!\cdots\!69}{51\!\cdots\!90}a^{16}-\frac{54\!\cdots\!42}{25\!\cdots\!45}a^{15}+\frac{59\!\cdots\!79}{10\!\cdots\!78}a^{14}+\frac{27\!\cdots\!19}{51\!\cdots\!90}a^{13}-\frac{27\!\cdots\!16}{51\!\cdots\!89}a^{12}+\frac{85\!\cdots\!13}{51\!\cdots\!90}a^{11}-\frac{56\!\cdots\!71}{51\!\cdots\!90}a^{10}-\frac{49\!\cdots\!31}{10\!\cdots\!78}a^{9}-\frac{15\!\cdots\!51}{37\!\cdots\!35}a^{8}+\frac{71\!\cdots\!11}{51\!\cdots\!90}a^{7}-\frac{21\!\cdots\!63}{98\!\cdots\!30}a^{6}-\frac{62\!\cdots\!96}{25\!\cdots\!45}a^{5}+\frac{81\!\cdots\!37}{51\!\cdots\!90}a^{4}+\frac{62\!\cdots\!39}{51\!\cdots\!90}a^{3}-\frac{74\!\cdots\!01}{53\!\cdots\!05}a^{2}+\frac{15\!\cdots\!48}{37\!\cdots\!35}a+\frac{67\!\cdots\!97}{10\!\cdots\!61}$, $\frac{17\!\cdots\!63}{37\!\cdots\!35}a^{17}-\frac{15\!\cdots\!37}{74\!\cdots\!70}a^{16}-\frac{12\!\cdots\!77}{74\!\cdots\!70}a^{15}+\frac{10\!\cdots\!49}{74\!\cdots\!70}a^{14}-\frac{92\!\cdots\!67}{74\!\cdots\!70}a^{13}-\frac{20\!\cdots\!76}{37\!\cdots\!35}a^{12}+\frac{11\!\cdots\!21}{37\!\cdots\!35}a^{11}-\frac{86\!\cdots\!33}{14\!\cdots\!54}a^{10}-\frac{54\!\cdots\!03}{10\!\cdots\!10}a^{9}-\frac{24\!\cdots\!62}{37\!\cdots\!35}a^{8}+\frac{18\!\cdots\!88}{53\!\cdots\!05}a^{7}-\frac{52\!\cdots\!27}{14\!\cdots\!90}a^{6}-\frac{19\!\cdots\!11}{74\!\cdots\!70}a^{5}+\frac{51\!\cdots\!71}{74\!\cdots\!70}a^{4}-\frac{26\!\cdots\!01}{10\!\cdots\!10}a^{3}-\frac{10\!\cdots\!26}{37\!\cdots\!35}a^{2}+\frac{22\!\cdots\!13}{53\!\cdots\!05}a-\frac{22\!\cdots\!60}{10\!\cdots\!61}$
| 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: | \( 21541865.688527487 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 21541865.688527487 \cdot 3}{6\cdot\sqrt{21423624263414745793258586112}}\cr\approx \mathstrut & 1.12311870918895 \end{aligned}\] (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 + 3*x^16 + 36*x^15 - 72*x^14 + 24*x^13 + 677*x^12 - 2154*x^11 + 650*x^10 - 12520*x^9 + 93111*x^8 - 184794*x^7 + 51751*x^6 + 253950*x^5 - 276628*x^4 - 32494*x^3 + 217588*x^2 - 143080*x + 34300)
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 + 3*x^16 + 36*x^15 - 72*x^14 + 24*x^13 + 677*x^12 - 2154*x^11 + 650*x^10 - 12520*x^9 + 93111*x^8 - 184794*x^7 + 51751*x^6 + 253950*x^5 - 276628*x^4 - 32494*x^3 + 217588*x^2 - 143080*x + 34300, 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 + 3*x^16 + 36*x^15 - 72*x^14 + 24*x^13 + 677*x^12 - 2154*x^11 + 650*x^10 - 12520*x^9 + 93111*x^8 - 184794*x^7 + 51751*x^6 + 253950*x^5 - 276628*x^4 - 32494*x^3 + 217588*x^2 - 143080*x + 34300);
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 + 3*x^16 + 36*x^15 - 72*x^14 + 24*x^13 + 677*x^12 - 2154*x^11 + 650*x^10 - 12520*x^9 + 93111*x^8 - 184794*x^7 + 51751*x^6 + 253950*x^5 - 276628*x^4 - 32494*x^3 + 217588*x^2 - 143080*x + 34300);
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
$C_3\times S_3^2$ (as 18T46):
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 108 |
| The 27 conjugacy class representatives for $C_3\times S_3^2$ |
| Character table for $C_3\times S_3^2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.728.1, 6.0.73008.1, 6.0.14309568.1 |
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 12 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.293442146934784486539264.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.2.0.1}{2} }^{9}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.4.2 | $x^{6} - 2 x^{3} + 4$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
| 2.12.22.27 | $x^{12} + 8 x^{9} + 16 x^{8} + 4 x^{7} + 2 x^{6} + 40 x^{5} + 104 x^{4} - 48 x^{3} - 12 x^{2} + 56 x + 196$ | $6$ | $2$ | $22$ | $C_6\times S_3$ | $[3]_{3}^{6}$ | |
|
\(3\)
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.6.3.1 | $x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(13\)
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 13.3.2.3 | $x^{3} + 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.6.3.1 | $x^{6} + 390 x^{5} + 50743 x^{4} + 2208202 x^{3} + 765301 x^{2} + 5017316 x + 24555184$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 13.6.5.1 | $x^{6} + 52$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |