Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 90*x^9 + 115*x^8 + 2349*x^7 - 943*x^6 - 26327*x^5 - 21284*x^4 + 102168*x^3 + 217794*x^2 + 148930*x + 30647)
gp: K = bnfinit(y^11 - y^10 - 90*y^9 + 115*y^8 + 2349*y^7 - 943*y^6 - 26327*y^5 - 21284*y^4 + 102168*y^3 + 217794*y^2 + 148930*y + 30647, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - x^10 - 90*x^9 + 115*x^8 + 2349*x^7 - 943*x^6 - 26327*x^5 - 21284*x^4 + 102168*x^3 + 217794*x^2 + 148930*x + 30647);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^11 - x^10 - 90*x^9 + 115*x^8 + 2349*x^7 - 943*x^6 - 26327*x^5 - 21284*x^4 + 102168*x^3 + 217794*x^2 + 148930*x + 30647)
\( x^{11} - x^{10} - 90 x^{9} + 115 x^{8} + 2349 x^{7} - 943 x^{6} - 26327 x^{5} - 21284 x^{4} + \cdots + 30647 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[11, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(97393677359695041798001\)
\(\medspace = 199^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(122.99\) | 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: | $199^{10/11}\approx 122.98904608708088$ | ||
| Ramified primes: |
\(199\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $11$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(199\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{199}(1,·)$, $\chi_{199}(103,·)$, $\chi_{199}(139,·)$, $\chi_{199}(114,·)$, $\chi_{199}(61,·)$, $\chi_{199}(18,·)$, $\chi_{199}(121,·)$, $\chi_{199}(188,·)$, $\chi_{199}(125,·)$, $\chi_{199}(62,·)$, $\chi_{199}(63,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19}a^{9}-\frac{2}{19}a^{8}-\frac{4}{19}a^{6}-\frac{3}{19}a^{5}+\frac{9}{19}a^{3}+\frac{6}{19}a^{2}-\frac{7}{19}a$, $\frac{1}{26\!\cdots\!03}a^{10}+\frac{16971015678963}{26\!\cdots\!03}a^{9}+\frac{543069253037908}{26\!\cdots\!03}a^{8}+\frac{12\!\cdots\!74}{26\!\cdots\!03}a^{7}+\frac{266849497773127}{26\!\cdots\!03}a^{6}+\frac{175733922112155}{26\!\cdots\!03}a^{5}-\frac{398340715614229}{26\!\cdots\!03}a^{4}-\frac{176975995579611}{26\!\cdots\!03}a^{3}+\frac{320511903583959}{26\!\cdots\!03}a^{2}-\frac{11\!\cdots\!53}{26\!\cdots\!03}a-\frac{16532606958192}{138218755975837}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| 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: |
$\frac{94264464873978}{26\!\cdots\!03}a^{10}-\frac{263592564201900}{26\!\cdots\!03}a^{9}-\frac{80\!\cdots\!78}{26\!\cdots\!03}a^{8}+\frac{25\!\cdots\!18}{26\!\cdots\!03}a^{7}+\frac{92\!\cdots\!54}{138218755975837}a^{6}-\frac{40\!\cdots\!65}{26\!\cdots\!03}a^{5}-\frac{17\!\cdots\!42}{26\!\cdots\!03}a^{4}+\frac{11\!\cdots\!23}{26\!\cdots\!03}a^{3}+\frac{75\!\cdots\!80}{26\!\cdots\!03}a^{2}+\frac{69\!\cdots\!56}{26\!\cdots\!03}a+\frac{85\!\cdots\!44}{138218755975837}$, $\frac{1257291276552}{26\!\cdots\!03}a^{10}-\frac{1861218622908}{26\!\cdots\!03}a^{9}-\frac{75568451629225}{26\!\cdots\!03}a^{8}+\frac{196166342996736}{26\!\cdots\!03}a^{7}-\frac{112972919140368}{26\!\cdots\!03}a^{6}-\frac{953827826280281}{26\!\cdots\!03}a^{5}+\frac{28\!\cdots\!63}{26\!\cdots\!03}a^{4}+\frac{10\!\cdots\!60}{26\!\cdots\!03}a^{3}-\frac{24\!\cdots\!36}{26\!\cdots\!03}a^{2}-\frac{42\!\cdots\!20}{26\!\cdots\!03}a-\frac{10\!\cdots\!22}{138218755975837}$, $\frac{53869341617323}{26\!\cdots\!03}a^{10}-\frac{113246568754219}{26\!\cdots\!03}a^{9}-\frac{47\!\cdots\!31}{26\!\cdots\!03}a^{8}+\frac{11\!\cdots\!76}{26\!\cdots\!03}a^{7}+\frac{11\!\cdots\!43}{26\!\cdots\!03}a^{6}-\frac{17\!\cdots\!79}{26\!\cdots\!03}a^{5}-\frac{12\!\cdots\!67}{26\!\cdots\!03}a^{4}+\frac{22\!\cdots\!10}{26\!\cdots\!03}a^{3}+\frac{52\!\cdots\!85}{26\!\cdots\!03}a^{2}+\frac{57\!\cdots\!77}{26\!\cdots\!03}a+\frac{76\!\cdots\!16}{138218755975837}$, $\frac{27931986006281}{26\!\cdots\!03}a^{10}-\frac{74182248301795}{26\!\cdots\!03}a^{9}-\frac{23\!\cdots\!97}{26\!\cdots\!03}a^{8}+\frac{71\!\cdots\!25}{26\!\cdots\!03}a^{7}+\frac{49\!\cdots\!00}{26\!\cdots\!03}a^{6}-\frac{11\!\cdots\!87}{26\!\cdots\!03}a^{5}-\frac{46\!\cdots\!84}{26\!\cdots\!03}a^{4}+\frac{29\!\cdots\!27}{26\!\cdots\!03}a^{3}+\frac{19\!\cdots\!87}{26\!\cdots\!03}a^{2}+\frac{17\!\cdots\!48}{26\!\cdots\!03}a+\frac{20\!\cdots\!19}{138218755975837}$, $\frac{4407142544885}{26\!\cdots\!03}a^{10}-\frac{722575824500}{138218755975837}a^{9}-\frac{367359857212044}{26\!\cdots\!03}a^{8}+\frac{12\!\cdots\!62}{26\!\cdots\!03}a^{7}+\frac{75\!\cdots\!61}{26\!\cdots\!03}a^{6}-\frac{20\!\cdots\!78}{26\!\cdots\!03}a^{5}-\frac{71\!\cdots\!88}{26\!\cdots\!03}a^{4}+\frac{58\!\cdots\!08}{26\!\cdots\!03}a^{3}+\frac{31\!\cdots\!84}{26\!\cdots\!03}a^{2}+\frac{28\!\cdots\!90}{26\!\cdots\!03}a+\frac{36\!\cdots\!54}{138218755975837}$, $\frac{4196360430830}{26\!\cdots\!03}a^{10}-\frac{11914193832232}{26\!\cdots\!03}a^{9}-\frac{365708276424320}{26\!\cdots\!03}a^{8}+\frac{11\!\cdots\!96}{26\!\cdots\!03}a^{7}+\frac{85\!\cdots\!03}{26\!\cdots\!03}a^{6}-\frac{19\!\cdots\!56}{26\!\cdots\!03}a^{5}-\frac{92\!\cdots\!01}{26\!\cdots\!03}a^{4}+\frac{56\!\cdots\!39}{26\!\cdots\!03}a^{3}+\frac{22\!\cdots\!59}{138218755975837}a^{2}+\frac{41\!\cdots\!89}{26\!\cdots\!03}a+\frac{52\!\cdots\!37}{138218755975837}$, $\frac{4108679916148}{26\!\cdots\!03}a^{10}-\frac{10709416354207}{26\!\cdots\!03}a^{9}-\frac{352725496503344}{26\!\cdots\!03}a^{8}+\frac{10\!\cdots\!53}{26\!\cdots\!03}a^{7}+\frac{80\!\cdots\!63}{26\!\cdots\!03}a^{6}-\frac{17\!\cdots\!19}{26\!\cdots\!03}a^{5}-\frac{84\!\cdots\!16}{26\!\cdots\!03}a^{4}+\frac{46\!\cdots\!74}{26\!\cdots\!03}a^{3}+\frac{38\!\cdots\!88}{26\!\cdots\!03}a^{2}+\frac{36\!\cdots\!06}{26\!\cdots\!03}a+\frac{46\!\cdots\!99}{138218755975837}$, $\frac{38654294444635}{26\!\cdots\!03}a^{10}-\frac{5607895833474}{138218755975837}a^{9}-\frac{32\!\cdots\!54}{26\!\cdots\!03}a^{8}+\frac{10\!\cdots\!55}{26\!\cdots\!03}a^{7}+\frac{72\!\cdots\!54}{26\!\cdots\!03}a^{6}-\frac{16\!\cdots\!20}{26\!\cdots\!03}a^{5}-\frac{73\!\cdots\!27}{26\!\cdots\!03}a^{4}+\frac{45\!\cdots\!89}{26\!\cdots\!03}a^{3}+\frac{31\!\cdots\!12}{26\!\cdots\!03}a^{2}+\frac{29\!\cdots\!49}{26\!\cdots\!03}a+\frac{36\!\cdots\!99}{138218755975837}$, $\frac{34152357447652}{26\!\cdots\!03}a^{10}-\frac{90839613097357}{26\!\cdots\!03}a^{9}-\frac{29\!\cdots\!60}{26\!\cdots\!03}a^{8}+\frac{87\!\cdots\!68}{26\!\cdots\!03}a^{7}+\frac{65\!\cdots\!11}{26\!\cdots\!03}a^{6}-\frac{13\!\cdots\!57}{26\!\cdots\!03}a^{5}-\frac{66\!\cdots\!07}{26\!\cdots\!03}a^{4}+\frac{35\!\cdots\!23}{26\!\cdots\!03}a^{3}+\frac{28\!\cdots\!06}{26\!\cdots\!03}a^{2}+\frac{27\!\cdots\!07}{26\!\cdots\!03}a+\frac{38\!\cdots\!52}{138218755975837}$, $\frac{6833505285136}{26\!\cdots\!03}a^{10}-\frac{9408895992462}{26\!\cdots\!03}a^{9}-\frac{612109467717208}{26\!\cdots\!03}a^{8}+\frac{10\!\cdots\!94}{26\!\cdots\!03}a^{7}+\frac{15\!\cdots\!66}{26\!\cdots\!03}a^{6}-\frac{12\!\cdots\!48}{26\!\cdots\!03}a^{5}-\frac{17\!\cdots\!30}{26\!\cdots\!03}a^{4}-\frac{80\!\cdots\!08}{26\!\cdots\!03}a^{3}+\frac{74\!\cdots\!50}{26\!\cdots\!03}a^{2}+\frac{12\!\cdots\!73}{26\!\cdots\!03}a+\frac{30\!\cdots\!79}{138218755975837}$
| 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: | \( 114390451.519 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 114390451.519 \cdot 1}{2\cdot\sqrt{97393677359695041798001}}\cr\approx \mathstrut & 0.375339567150 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 90*x^9 + 115*x^8 + 2349*x^7 - 943*x^6 - 26327*x^5 - 21284*x^4 + 102168*x^3 + 217794*x^2 + 148930*x + 30647)
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^11 - x^10 - 90*x^9 + 115*x^8 + 2349*x^7 - 943*x^6 - 26327*x^5 - 21284*x^4 + 102168*x^3 + 217794*x^2 + 148930*x + 30647, 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^11 - x^10 - 90*x^9 + 115*x^8 + 2349*x^7 - 943*x^6 - 26327*x^5 - 21284*x^4 + 102168*x^3 + 217794*x^2 + 148930*x + 30647);
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^11 - x^10 - 90*x^9 + 115*x^8 + 2349*x^7 - 943*x^6 - 26327*x^5 - 21284*x^4 + 102168*x^3 + 217794*x^2 + 148930*x + 30647);
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 cyclic group of order 11 |
| The 11 conjugacy class representatives for $C_{11}$ |
| Character table for $C_{11}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.11.0.1}{11} }$ | ${\href{/padicField/3.11.0.1}{11} }$ | ${\href{/padicField/5.11.0.1}{11} }$ | ${\href{/padicField/7.11.0.1}{11} }$ | ${\href{/padicField/11.11.0.1}{11} }$ | ${\href{/padicField/13.11.0.1}{11} }$ | ${\href{/padicField/17.11.0.1}{11} }$ | ${\href{/padicField/19.1.0.1}{1} }^{11}$ | ${\href{/padicField/23.11.0.1}{11} }$ | ${\href{/padicField/29.11.0.1}{11} }$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.1.0.1}{1} }^{11}$ | ${\href{/padicField/41.11.0.1}{11} }$ | ${\href{/padicField/43.1.0.1}{1} }^{11}$ | ${\href{/padicField/47.11.0.1}{11} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.11.0.1}{11} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(199\)
| 199.11.10.1 | $x^{11} + 199$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |