Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 180*x^9 + 13*x^8 + 11655*x^7 + 12159*x^6 - 316973*x^5 - 720142*x^4 + 2670510*x^3 + 10551746*x^2 + 10752776*x + 3098903)
gp: K = bnfinit(y^11 - y^10 - 180*y^9 + 13*y^8 + 11655*y^7 + 12159*y^6 - 316973*y^5 - 720142*y^4 + 2670510*y^3 + 10551746*y^2 + 10752776*y + 3098903, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - x^10 - 180*x^9 + 13*x^8 + 11655*x^7 + 12159*x^6 - 316973*x^5 - 720142*x^4 + 2670510*x^3 + 10551746*x^2 + 10752776*x + 3098903);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^11 - x^10 - 180*x^9 + 13*x^8 + 11655*x^7 + 12159*x^6 - 316973*x^5 - 720142*x^4 + 2670510*x^3 + 10551746*x^2 + 10752776*x + 3098903)
\( x^{11} - x^{10} - 180 x^{9} + 13 x^{8} + 11655 x^{7} + 12159 x^{6} - 316973 x^{5} - 720142 x^{4} + \cdots + 3098903 \)
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: |
\(97253461433805715000527049\)
\(\medspace = 397^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(230.43\) | 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: | $397^{10/11}\approx 230.42883510401091$ | ||
| Ramified primes: |
\(397\)
| 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: | \(397\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{397}(256,·)$, $\chi_{397}(1,·)$, $\chi_{397}(290,·)$, $\chi_{397}(99,·)$, $\chi_{397}(167,·)$, $\chi_{397}(393,·)$, $\chi_{397}(333,·)$, $\chi_{397}(16,·)$, $\chi_{397}(273,·)$, $\chi_{397}(126,·)$, $\chi_{397}(31,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{593}a^{9}-\frac{62}{593}a^{8}-\frac{216}{593}a^{7}+\frac{252}{593}a^{6}+\frac{259}{593}a^{5}+\frac{221}{593}a^{4}+\frac{109}{593}a^{3}+\frac{288}{593}a^{2}-\frac{16}{593}a+\frac{125}{593}$, $\frac{1}{34\!\cdots\!81}a^{10}-\frac{18\!\cdots\!64}{34\!\cdots\!81}a^{9}-\frac{15\!\cdots\!75}{34\!\cdots\!81}a^{8}-\frac{88\!\cdots\!40}{34\!\cdots\!81}a^{7}+\frac{10\!\cdots\!87}{34\!\cdots\!81}a^{6}-\frac{20\!\cdots\!65}{34\!\cdots\!81}a^{5}-\frac{91\!\cdots\!45}{34\!\cdots\!81}a^{4}+\frac{15\!\cdots\!30}{34\!\cdots\!81}a^{3}-\frac{28\!\cdots\!35}{34\!\cdots\!81}a^{2}-\frac{14\!\cdots\!08}{34\!\cdots\!81}a+\frac{34\!\cdots\!18}{83\!\cdots\!41}$
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: |
$\frac{22\!\cdots\!16}{34\!\cdots\!81}a^{10}-\frac{20\!\cdots\!66}{34\!\cdots\!81}a^{9}-\frac{36\!\cdots\!38}{34\!\cdots\!81}a^{8}+\frac{27\!\cdots\!44}{34\!\cdots\!81}a^{7}+\frac{21\!\cdots\!76}{34\!\cdots\!81}a^{6}-\frac{10\!\cdots\!64}{34\!\cdots\!81}a^{5}-\frac{59\!\cdots\!77}{34\!\cdots\!81}a^{4}+\frac{10\!\cdots\!11}{34\!\cdots\!81}a^{3}+\frac{63\!\cdots\!37}{34\!\cdots\!81}a^{2}+\frac{64\!\cdots\!09}{34\!\cdots\!81}a+\frac{33\!\cdots\!05}{83\!\cdots\!41}$, $\frac{91\!\cdots\!17}{34\!\cdots\!81}a^{10}-\frac{42\!\cdots\!35}{34\!\cdots\!81}a^{9}-\frac{14\!\cdots\!77}{34\!\cdots\!81}a^{8}+\frac{54\!\cdots\!05}{34\!\cdots\!81}a^{7}+\frac{77\!\cdots\!99}{34\!\cdots\!81}a^{6}-\frac{19\!\cdots\!83}{34\!\cdots\!81}a^{5}-\frac{17\!\cdots\!28}{34\!\cdots\!81}a^{4}+\frac{11\!\cdots\!27}{34\!\cdots\!81}a^{3}+\frac{15\!\cdots\!40}{34\!\cdots\!81}a^{2}+\frac{19\!\cdots\!13}{34\!\cdots\!81}a+\frac{14\!\cdots\!59}{83\!\cdots\!41}$, $\frac{52\!\cdots\!57}{34\!\cdots\!81}a^{10}+\frac{11\!\cdots\!43}{34\!\cdots\!81}a^{9}-\frac{49\!\cdots\!02}{34\!\cdots\!81}a^{8}-\frac{16\!\cdots\!22}{34\!\cdots\!81}a^{7}-\frac{86\!\cdots\!05}{34\!\cdots\!81}a^{6}+\frac{73\!\cdots\!96}{34\!\cdots\!81}a^{5}+\frac{17\!\cdots\!75}{34\!\cdots\!81}a^{4}-\frac{82\!\cdots\!72}{34\!\cdots\!81}a^{3}-\frac{33\!\cdots\!60}{34\!\cdots\!81}a^{2}-\frac{35\!\cdots\!35}{34\!\cdots\!81}a-\frac{24\!\cdots\!43}{83\!\cdots\!41}$, $\frac{16\!\cdots\!51}{34\!\cdots\!81}a^{10}-\frac{52\!\cdots\!68}{34\!\cdots\!81}a^{9}-\frac{28\!\cdots\!47}{34\!\cdots\!81}a^{8}+\frac{66\!\cdots\!20}{34\!\cdots\!81}a^{7}+\frac{17\!\cdots\!49}{34\!\cdots\!81}a^{6}-\frac{20\!\cdots\!51}{34\!\cdots\!81}a^{5}-\frac{46\!\cdots\!63}{34\!\cdots\!81}a^{4}-\frac{52\!\cdots\!46}{34\!\cdots\!81}a^{3}+\frac{44\!\cdots\!35}{34\!\cdots\!81}a^{2}+\frac{60\!\cdots\!26}{34\!\cdots\!81}a+\frac{48\!\cdots\!52}{83\!\cdots\!41}$, $\frac{17\!\cdots\!03}{34\!\cdots\!81}a^{10}+\frac{86\!\cdots\!78}{34\!\cdots\!81}a^{9}-\frac{24\!\cdots\!99}{34\!\cdots\!81}a^{8}-\frac{12\!\cdots\!46}{34\!\cdots\!81}a^{7}+\frac{10\!\cdots\!09}{34\!\cdots\!81}a^{6}+\frac{62\!\cdots\!50}{34\!\cdots\!81}a^{5}-\frac{93\!\cdots\!46}{34\!\cdots\!81}a^{4}-\frac{95\!\cdots\!94}{34\!\cdots\!81}a^{3}-\frac{98\!\cdots\!84}{34\!\cdots\!81}a^{2}+\frac{13\!\cdots\!48}{34\!\cdots\!81}a+\frac{40\!\cdots\!59}{83\!\cdots\!41}$, $\frac{53\!\cdots\!72}{34\!\cdots\!81}a^{10}-\frac{19\!\cdots\!67}{34\!\cdots\!81}a^{9}-\frac{90\!\cdots\!95}{34\!\cdots\!81}a^{8}+\frac{24\!\cdots\!63}{34\!\cdots\!81}a^{7}+\frac{55\!\cdots\!28}{34\!\cdots\!81}a^{6}-\frac{85\!\cdots\!40}{34\!\cdots\!81}a^{5}-\frac{14\!\cdots\!51}{34\!\cdots\!81}a^{4}+\frac{27\!\cdots\!33}{34\!\cdots\!81}a^{3}+\frac{14\!\cdots\!54}{34\!\cdots\!81}a^{2}+\frac{15\!\cdots\!29}{34\!\cdots\!81}a+\frac{38\!\cdots\!49}{83\!\cdots\!41}$, $\frac{85\!\cdots\!00}{34\!\cdots\!81}a^{10}+\frac{33\!\cdots\!27}{34\!\cdots\!81}a^{9}-\frac{12\!\cdots\!38}{34\!\cdots\!81}a^{8}-\frac{48\!\cdots\!10}{34\!\cdots\!81}a^{7}+\frac{55\!\cdots\!96}{34\!\cdots\!81}a^{6}+\frac{23\!\cdots\!47}{34\!\cdots\!81}a^{5}-\frac{82\!\cdots\!08}{34\!\cdots\!81}a^{4}-\frac{42\!\cdots\!24}{34\!\cdots\!81}a^{3}+\frac{10\!\cdots\!19}{34\!\cdots\!81}a^{2}+\frac{18\!\cdots\!14}{34\!\cdots\!81}a+\frac{36\!\cdots\!79}{83\!\cdots\!41}$, $\frac{39\!\cdots\!12}{34\!\cdots\!81}a^{10}-\frac{12\!\cdots\!33}{34\!\cdots\!81}a^{9}-\frac{66\!\cdots\!24}{34\!\cdots\!81}a^{8}+\frac{15\!\cdots\!11}{34\!\cdots\!81}a^{7}+\frac{39\!\cdots\!49}{34\!\cdots\!81}a^{6}-\frac{49\!\cdots\!31}{34\!\cdots\!81}a^{5}-\frac{10\!\cdots\!86}{34\!\cdots\!81}a^{4}-\frac{73\!\cdots\!38}{34\!\cdots\!81}a^{3}+\frac{96\!\cdots\!25}{34\!\cdots\!81}a^{2}+\frac{12\!\cdots\!20}{34\!\cdots\!81}a+\frac{10\!\cdots\!40}{83\!\cdots\!41}$, $\frac{35\!\cdots\!06}{34\!\cdots\!81}a^{10}-\frac{81\!\cdots\!12}{34\!\cdots\!81}a^{9}-\frac{80\!\cdots\!12}{34\!\cdots\!81}a^{8}+\frac{10\!\cdots\!53}{34\!\cdots\!81}a^{7}+\frac{65\!\cdots\!51}{34\!\cdots\!81}a^{6}-\frac{40\!\cdots\!01}{34\!\cdots\!81}a^{5}-\frac{22\!\cdots\!61}{34\!\cdots\!81}a^{4}+\frac{31\!\cdots\!33}{34\!\cdots\!81}a^{3}+\frac{25\!\cdots\!82}{34\!\cdots\!81}a^{2}+\frac{31\!\cdots\!77}{34\!\cdots\!81}a+\frac{23\!\cdots\!40}{83\!\cdots\!41}$, $\frac{10\!\cdots\!12}{34\!\cdots\!81}a^{10}-\frac{21\!\cdots\!84}{34\!\cdots\!81}a^{9}-\frac{17\!\cdots\!40}{34\!\cdots\!81}a^{8}+\frac{21\!\cdots\!50}{34\!\cdots\!81}a^{7}+\frac{11\!\cdots\!44}{34\!\cdots\!81}a^{6}-\frac{14\!\cdots\!29}{34\!\cdots\!81}a^{5}-\frac{29\!\cdots\!35}{34\!\cdots\!81}a^{4}-\frac{30\!\cdots\!75}{34\!\cdots\!81}a^{3}+\frac{27\!\cdots\!62}{34\!\cdots\!81}a^{2}+\frac{53\!\cdots\!58}{34\!\cdots\!81}a+\frac{48\!\cdots\!35}{83\!\cdots\!41}$
| 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: | \( 1592800927.62 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 1592800927.62 \cdot 1}{2\cdot\sqrt{97253461433805715000527049}}\cr\approx \mathstrut & 0.165389876449 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 180*x^9 + 13*x^8 + 11655*x^7 + 12159*x^6 - 316973*x^5 - 720142*x^4 + 2670510*x^3 + 10551746*x^2 + 10752776*x + 3098903)
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 - 180*x^9 + 13*x^8 + 11655*x^7 + 12159*x^6 - 316973*x^5 - 720142*x^4 + 2670510*x^3 + 10551746*x^2 + 10752776*x + 3098903, 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 - 180*x^9 + 13*x^8 + 11655*x^7 + 12159*x^6 - 316973*x^5 - 720142*x^4 + 2670510*x^3 + 10551746*x^2 + 10752776*x + 3098903);
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 - 180*x^9 + 13*x^8 + 11655*x^7 + 12159*x^6 - 316973*x^5 - 720142*x^4 + 2670510*x^3 + 10551746*x^2 + 10752776*x + 3098903);
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 | ${\href{/padicField/2.11.0.1}{11} }$ | ${\href{/padicField/3.11.0.1}{11} }$ | ${\href{/padicField/5.11.0.1}{11} }$ | ${\href{/padicField/7.11.0.1}{11} }$ | ${\href{/padicField/11.11.0.1}{11} }$ | ${\href{/padicField/13.11.0.1}{11} }$ | ${\href{/padicField/17.11.0.1}{11} }$ | ${\href{/padicField/19.11.0.1}{11} }$ | ${\href{/padicField/23.11.0.1}{11} }$ | ${\href{/padicField/29.11.0.1}{11} }$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.11.0.1}{11} }$ | ${\href{/padicField/41.1.0.1}{1} }^{11}$ | ${\href{/padicField/43.11.0.1}{11} }$ | ${\href{/padicField/47.11.0.1}{11} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.11.0.1}{11} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(397\)
| Deg $11$ | $11$ | $1$ | $10$ |