Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 19*x^16 - 46*x^15 + 209*x^14 - 457*x^13 + 1289*x^12 - 2046*x^11 + 4263*x^10 - 5541*x^9 + 9392*x^8 - 9256*x^7 + 11934*x^6 - 8321*x^5 + 9457*x^4 - 4564*x^3 + 3678*x^2 + 60*x + 1)
gp: K = bnfinit(y^18 - 3*y^17 + 19*y^16 - 46*y^15 + 209*y^14 - 457*y^13 + 1289*y^12 - 2046*y^11 + 4263*y^10 - 5541*y^9 + 9392*y^8 - 9256*y^7 + 11934*y^6 - 8321*y^5 + 9457*y^4 - 4564*y^3 + 3678*y^2 + 60*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 + 19*x^16 - 46*x^15 + 209*x^14 - 457*x^13 + 1289*x^12 - 2046*x^11 + 4263*x^10 - 5541*x^9 + 9392*x^8 - 9256*x^7 + 11934*x^6 - 8321*x^5 + 9457*x^4 - 4564*x^3 + 3678*x^2 + 60*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 + 19*x^16 - 46*x^15 + 209*x^14 - 457*x^13 + 1289*x^12 - 2046*x^11 + 4263*x^10 - 5541*x^9 + 9392*x^8 - 9256*x^7 + 11934*x^6 - 8321*x^5 + 9457*x^4 - 4564*x^3 + 3678*x^2 + 60*x + 1)
\( x^{18} - 3 x^{17} + 19 x^{16} - 46 x^{15} + 209 x^{14} - 457 x^{13} + 1289 x^{12} - 2046 x^{11} + \cdots + 1 \)
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: |
\(-51956851764606118870683564963\)
\(\medspace = -\,3^{9}\cdot 1129^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(39.38\) | 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: | $3^{1/2}1129^{1/2}\approx 58.19793810780585$ | ||
| Ramified primes: |
\(3\), \(1129\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19}a^{15}-\frac{4}{19}a^{14}+\frac{1}{19}a^{13}+\frac{2}{19}a^{12}-\frac{1}{19}a^{11}-\frac{7}{19}a^{10}+\frac{5}{19}a^{9}+\frac{4}{19}a^{8}-\frac{5}{19}a^{7}-\frac{5}{19}a^{6}+\frac{3}{19}a^{5}-\frac{5}{19}a^{4}+\frac{3}{19}a^{3}-\frac{9}{19}a^{2}+\frac{1}{19}$, $\frac{1}{19}a^{16}+\frac{4}{19}a^{14}+\frac{6}{19}a^{13}+\frac{7}{19}a^{12}+\frac{8}{19}a^{11}-\frac{4}{19}a^{10}+\frac{5}{19}a^{9}-\frac{8}{19}a^{8}-\frac{6}{19}a^{7}+\frac{2}{19}a^{6}+\frac{7}{19}a^{5}+\frac{2}{19}a^{4}+\frac{3}{19}a^{3}+\frac{2}{19}a^{2}+\frac{1}{19}a+\frac{4}{19}$, $\frac{1}{17\!\cdots\!91}a^{17}+\frac{49\!\cdots\!36}{17\!\cdots\!91}a^{16}+\frac{72\!\cdots\!13}{61\!\cdots\!79}a^{15}-\frac{53\!\cdots\!62}{17\!\cdots\!91}a^{14}-\frac{69\!\cdots\!85}{17\!\cdots\!91}a^{13}+\frac{15\!\cdots\!80}{17\!\cdots\!91}a^{12}-\frac{13\!\cdots\!08}{17\!\cdots\!91}a^{11}-\frac{39\!\cdots\!32}{17\!\cdots\!91}a^{10}+\frac{14\!\cdots\!29}{17\!\cdots\!91}a^{9}-\frac{36\!\cdots\!81}{17\!\cdots\!91}a^{8}-\frac{34\!\cdots\!61}{93\!\cdots\!89}a^{7}+\frac{51\!\cdots\!82}{17\!\cdots\!91}a^{6}-\frac{41\!\cdots\!23}{17\!\cdots\!91}a^{5}+\frac{30\!\cdots\!68}{17\!\cdots\!91}a^{4}-\frac{26\!\cdots\!41}{93\!\cdots\!89}a^{3}+\frac{47\!\cdots\!01}{17\!\cdots\!91}a^{2}-\frac{87\!\cdots\!15}{17\!\cdots\!91}a+\frac{13\!\cdots\!88}{17\!\cdots\!91}$
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
$C_{2}\times C_{38}$, which has order $76$ (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: |
\( -\frac{29333510837476704074339}{1780344773928222178369891} a^{17} + \frac{88213177518867648875147}{1780344773928222178369891} a^{16} - \frac{19222870878466126699084}{61391199100973178564479} a^{15} + \frac{1351885121027244340292883}{1780344773928222178369891} a^{14} - \frac{6131174220851336992783571}{1780344773928222178369891} a^{13} + \frac{13428241015122774316752373}{1780344773928222178369891} a^{12} - \frac{37809088739123548025245491}{1780344773928222178369891} a^{11} + \frac{60082497022956651666520939}{1780344773928222178369891} a^{10} - \frac{124909412125943900374206092}{1780344773928222178369891} a^{9} + \frac{162620973139084496762883691}{1780344773928222178369891} a^{8} - \frac{14474050186287256946169357}{93702356522538009387889} a^{7} + \frac{271601669308298606333765854}{1780344773928222178369891} a^{6} - \frac{348723085871212613963280841}{1780344773928222178369891} a^{5} + \frac{244123161069096676601877974}{1780344773928222178369891} a^{4} - \frac{14530595680273679248173192}{93702356522538009387889} a^{3} + \frac{135441400744220539858470835}{1780344773928222178369891} a^{2} - \frac{106849345692613212166805757}{1780344773928222178369891} a + \frac{37294591531552648308581}{1780344773928222178369891} \)
(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: |
$\frac{21\!\cdots\!27}{17\!\cdots\!91}a^{17}-\frac{59\!\cdots\!78}{17\!\cdots\!91}a^{16}+\frac{14\!\cdots\!11}{61\!\cdots\!79}a^{15}-\frac{92\!\cdots\!48}{17\!\cdots\!91}a^{14}+\frac{45\!\cdots\!16}{17\!\cdots\!91}a^{13}-\frac{92\!\cdots\!74}{17\!\cdots\!91}a^{12}+\frac{27\!\cdots\!52}{17\!\cdots\!91}a^{11}-\frac{41\!\cdots\!25}{17\!\cdots\!91}a^{10}+\frac{94\!\cdots\!19}{17\!\cdots\!91}a^{9}-\frac{11\!\cdots\!14}{17\!\cdots\!91}a^{8}+\frac{11\!\cdots\!33}{93\!\cdots\!89}a^{7}-\frac{19\!\cdots\!70}{17\!\cdots\!91}a^{6}+\frac{28\!\cdots\!48}{17\!\cdots\!91}a^{5}-\frac{17\!\cdots\!19}{17\!\cdots\!91}a^{4}+\frac{12\!\cdots\!43}{93\!\cdots\!89}a^{3}-\frac{89\!\cdots\!34}{17\!\cdots\!91}a^{2}+\frac{10\!\cdots\!39}{17\!\cdots\!91}a+\frac{17\!\cdots\!86}{17\!\cdots\!91}$, $\frac{13\!\cdots\!37}{17\!\cdots\!91}a^{17}-\frac{41\!\cdots\!05}{17\!\cdots\!91}a^{16}+\frac{90\!\cdots\!32}{61\!\cdots\!79}a^{15}-\frac{66\!\cdots\!97}{17\!\cdots\!91}a^{14}+\frac{29\!\cdots\!06}{17\!\cdots\!91}a^{13}-\frac{67\!\cdots\!03}{17\!\cdots\!91}a^{12}+\frac{18\!\cdots\!57}{17\!\cdots\!91}a^{11}-\frac{32\!\cdots\!89}{17\!\cdots\!91}a^{10}+\frac{66\!\cdots\!24}{17\!\cdots\!91}a^{9}-\frac{91\!\cdots\!29}{17\!\cdots\!91}a^{8}+\frac{41\!\cdots\!38}{49\!\cdots\!31}a^{7}-\frac{16\!\cdots\!42}{17\!\cdots\!91}a^{6}+\frac{20\!\cdots\!71}{17\!\cdots\!91}a^{5}-\frac{15\!\cdots\!67}{17\!\cdots\!91}a^{4}+\frac{90\!\cdots\!00}{93\!\cdots\!89}a^{3}-\frac{88\!\cdots\!33}{17\!\cdots\!91}a^{2}+\frac{83\!\cdots\!38}{17\!\cdots\!91}a-\frac{24\!\cdots\!47}{17\!\cdots\!91}$, $\frac{15\!\cdots\!91}{17\!\cdots\!91}a^{17}-\frac{41\!\cdots\!59}{17\!\cdots\!91}a^{16}+\frac{96\!\cdots\!78}{61\!\cdots\!79}a^{15}-\frac{62\!\cdots\!56}{17\!\cdots\!91}a^{14}+\frac{30\!\cdots\!94}{17\!\cdots\!91}a^{13}-\frac{61\!\cdots\!31}{17\!\cdots\!91}a^{12}+\frac{17\!\cdots\!71}{17\!\cdots\!91}a^{11}-\frac{26\!\cdots\!96}{17\!\cdots\!91}a^{10}+\frac{56\!\cdots\!89}{17\!\cdots\!91}a^{9}-\frac{71\!\cdots\!02}{17\!\cdots\!91}a^{8}+\frac{64\!\cdots\!44}{93\!\cdots\!89}a^{7}-\frac{11\!\cdots\!43}{17\!\cdots\!91}a^{6}+\frac{15\!\cdots\!05}{17\!\cdots\!91}a^{5}-\frac{10\!\cdots\!20}{17\!\cdots\!91}a^{4}+\frac{32\!\cdots\!93}{49\!\cdots\!31}a^{3}-\frac{54\!\cdots\!82}{17\!\cdots\!91}a^{2}+\frac{44\!\cdots\!34}{17\!\cdots\!91}a-\frac{15\!\cdots\!12}{17\!\cdots\!91}$, $\frac{50\!\cdots\!23}{17\!\cdots\!91}a^{17}+\frac{87\!\cdots\!29}{17\!\cdots\!91}a^{16}-\frac{22\!\cdots\!10}{61\!\cdots\!79}a^{15}+\frac{12\!\cdots\!98}{17\!\cdots\!91}a^{14}-\frac{11\!\cdots\!40}{17\!\cdots\!91}a^{13}+\frac{12\!\cdots\!54}{17\!\cdots\!91}a^{12}-\frac{14\!\cdots\!43}{17\!\cdots\!91}a^{11}+\frac{57\!\cdots\!82}{17\!\cdots\!91}a^{10}-\frac{44\!\cdots\!64}{17\!\cdots\!91}a^{9}+\frac{15\!\cdots\!68}{17\!\cdots\!91}a^{8}-\frac{60\!\cdots\!87}{93\!\cdots\!89}a^{7}+\frac{25\!\cdots\!08}{17\!\cdots\!91}a^{6}-\frac{11\!\cdots\!77}{17\!\cdots\!91}a^{5}+\frac{22\!\cdots\!91}{17\!\cdots\!91}a^{4}-\frac{47\!\cdots\!08}{93\!\cdots\!89}a^{3}+\frac{12\!\cdots\!05}{17\!\cdots\!91}a^{2}+\frac{20\!\cdots\!09}{17\!\cdots\!91}a+\frac{70\!\cdots\!81}{17\!\cdots\!91}$, $\frac{13\!\cdots\!77}{10\!\cdots\!23}a^{17}-\frac{33\!\cdots\!62}{10\!\cdots\!23}a^{16}+\frac{80\!\cdots\!44}{36\!\cdots\!87}a^{15}-\frac{48\!\cdots\!27}{10\!\cdots\!23}a^{14}+\frac{24\!\cdots\!51}{10\!\cdots\!23}a^{13}-\frac{46\!\cdots\!75}{10\!\cdots\!23}a^{12}+\frac{13\!\cdots\!57}{10\!\cdots\!23}a^{11}-\frac{18\!\cdots\!98}{10\!\cdots\!23}a^{10}+\frac{41\!\cdots\!86}{10\!\cdots\!23}a^{9}-\frac{48\!\cdots\!99}{10\!\cdots\!23}a^{8}+\frac{44\!\cdots\!73}{55\!\cdots\!17}a^{7}-\frac{71\!\cdots\!64}{10\!\cdots\!23}a^{6}+\frac{93\!\cdots\!43}{10\!\cdots\!23}a^{5}-\frac{60\!\cdots\!96}{10\!\cdots\!23}a^{4}+\frac{35\!\cdots\!02}{55\!\cdots\!17}a^{3}-\frac{28\!\cdots\!91}{10\!\cdots\!23}a^{2}+\frac{18\!\cdots\!27}{10\!\cdots\!23}a-\frac{80\!\cdots\!53}{10\!\cdots\!23}$, $\frac{47\!\cdots\!43}{16\!\cdots\!81}a^{17}-\frac{28\!\cdots\!09}{16\!\cdots\!81}a^{16}+\frac{21\!\cdots\!95}{55\!\cdots\!89}a^{15}-\frac{27\!\cdots\!37}{16\!\cdots\!81}a^{14}+\frac{60\!\cdots\!65}{16\!\cdots\!81}a^{13}-\frac{16\!\cdots\!21}{16\!\cdots\!81}a^{12}+\frac{23\!\cdots\!52}{16\!\cdots\!81}a^{11}+\frac{12\!\cdots\!23}{16\!\cdots\!81}a^{10}+\frac{57\!\cdots\!91}{16\!\cdots\!81}a^{9}+\frac{59\!\cdots\!58}{16\!\cdots\!81}a^{8}+\frac{45\!\cdots\!08}{85\!\cdots\!99}a^{7}+\frac{19\!\cdots\!95}{16\!\cdots\!81}a^{6}+\frac{72\!\cdots\!63}{16\!\cdots\!81}a^{5}+\frac{25\!\cdots\!73}{16\!\cdots\!81}a^{4}+\frac{36\!\cdots\!95}{85\!\cdots\!99}a^{3}+\frac{22\!\cdots\!68}{16\!\cdots\!81}a^{2}+\frac{37\!\cdots\!76}{16\!\cdots\!81}a+\frac{31\!\cdots\!56}{16\!\cdots\!81}$, $\frac{14\!\cdots\!82}{16\!\cdots\!81}a^{17}-\frac{42\!\cdots\!22}{16\!\cdots\!81}a^{16}+\frac{92\!\cdots\!32}{55\!\cdots\!89}a^{15}-\frac{62\!\cdots\!26}{16\!\cdots\!81}a^{14}+\frac{28\!\cdots\!15}{16\!\cdots\!81}a^{13}-\frac{60\!\cdots\!70}{16\!\cdots\!81}a^{12}+\frac{17\!\cdots\!94}{16\!\cdots\!81}a^{11}-\frac{25\!\cdots\!50}{16\!\cdots\!81}a^{10}+\frac{53\!\cdots\!88}{16\!\cdots\!81}a^{9}-\frac{65\!\cdots\!42}{16\!\cdots\!81}a^{8}+\frac{59\!\cdots\!85}{85\!\cdots\!99}a^{7}-\frac{99\!\cdots\!92}{16\!\cdots\!81}a^{6}+\frac{12\!\cdots\!70}{16\!\cdots\!81}a^{5}-\frac{81\!\cdots\!37}{16\!\cdots\!81}a^{4}+\frac{49\!\cdots\!72}{85\!\cdots\!99}a^{3}-\frac{42\!\cdots\!82}{16\!\cdots\!81}a^{2}+\frac{21\!\cdots\!29}{16\!\cdots\!81}a-\frac{11\!\cdots\!94}{16\!\cdots\!81}$, $\frac{23\!\cdots\!80}{17\!\cdots\!91}a^{17}-\frac{28\!\cdots\!55}{17\!\cdots\!91}a^{16}+\frac{11\!\cdots\!99}{61\!\cdots\!79}a^{15}-\frac{32\!\cdots\!21}{17\!\cdots\!91}a^{14}+\frac{32\!\cdots\!81}{17\!\cdots\!91}a^{13}-\frac{26\!\cdots\!33}{17\!\cdots\!91}a^{12}+\frac{13\!\cdots\!92}{17\!\cdots\!91}a^{11}-\frac{10\!\cdots\!48}{17\!\cdots\!91}a^{10}+\frac{33\!\cdots\!68}{17\!\cdots\!91}a^{9}+\frac{10\!\cdots\!75}{17\!\cdots\!91}a^{8}+\frac{29\!\cdots\!13}{93\!\cdots\!89}a^{7}+\frac{61\!\cdots\!71}{17\!\cdots\!91}a^{6}+\frac{49\!\cdots\!69}{17\!\cdots\!91}a^{5}+\frac{97\!\cdots\!35}{17\!\cdots\!91}a^{4}+\frac{22\!\cdots\!52}{93\!\cdots\!89}a^{3}+\frac{97\!\cdots\!72}{17\!\cdots\!91}a^{2}+\frac{15\!\cdots\!77}{17\!\cdots\!91}a+\frac{36\!\cdots\!84}{17\!\cdots\!91}$
| 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: | \( 290542.983381 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 290542.983381 \cdot 76}{6\cdot\sqrt{51956851764606118870683564963}}\cr\approx \mathstrut & 0.246416698659 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 19*x^16 - 46*x^15 + 209*x^14 - 457*x^13 + 1289*x^12 - 2046*x^11 + 4263*x^10 - 5541*x^9 + 9392*x^8 - 9256*x^7 + 11934*x^6 - 8321*x^5 + 9457*x^4 - 4564*x^3 + 3678*x^2 + 60*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 - 3*x^17 + 19*x^16 - 46*x^15 + 209*x^14 - 457*x^13 + 1289*x^12 - 2046*x^11 + 4263*x^10 - 5541*x^9 + 9392*x^8 - 9256*x^7 + 11934*x^6 - 8321*x^5 + 9457*x^4 - 4564*x^3 + 3678*x^2 + 60*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 - 3*x^17 + 19*x^16 - 46*x^15 + 209*x^14 - 457*x^13 + 1289*x^12 - 2046*x^11 + 4263*x^10 - 5541*x^9 + 9392*x^8 - 9256*x^7 + 11934*x^6 - 8321*x^5 + 9457*x^4 - 4564*x^3 + 3678*x^2 + 60*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 - 3*x^17 + 19*x^16 - 46*x^15 + 209*x^14 - 457*x^13 + 1289*x^12 - 2046*x^11 + 4263*x^10 - 5541*x^9 + 9392*x^8 - 9256*x^7 + 11934*x^6 - 8321*x^5 + 9457*x^4 - 4564*x^3 + 3678*x^2 + 60*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
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 12 conjugacy class representatives for $D_{18}$ |
| Character table for $D_{18}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.3.1129.1, 6.0.34415307.1, 9.9.1624709678881.1 |
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 18 sibling: | deg 18 |
| 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 | $18$ | R | $18$ | ${\href{/padicField/7.9.0.1}{9} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{9}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $18$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | $18$ | $18$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 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 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(1129\)
| $\Q_{1129}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1129}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |