Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 450*x^9 + 393*x^8 + 65785*x^7 - 76121*x^6 - 3728249*x^5 + 6483322*x^4 + 89323866*x^3 - 205338716*x^2 - 745717454*x + 2032282741)
gp: K = bnfinit(y^11 - y^10 - 450*y^9 + 393*y^8 + 65785*y^7 - 76121*y^6 - 3728249*y^5 + 6483322*y^4 + 89323866*y^3 - 205338716*y^2 - 745717454*y + 2032282741, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - x^10 - 450*x^9 + 393*x^8 + 65785*x^7 - 76121*x^6 - 3728249*x^5 + 6483322*x^4 + 89323866*x^3 - 205338716*x^2 - 745717454*x + 2032282741);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^11 - x^10 - 450*x^9 + 393*x^8 + 65785*x^7 - 76121*x^6 - 3728249*x^5 + 6483322*x^4 + 89323866*x^3 - 205338716*x^2 - 745717454*x + 2032282741)
\( x^{11} - x^{10} - 450 x^{9} + 393 x^{8} + 65785 x^{7} - 76121 x^{6} - 3728249 x^{5} + 6483322 x^{4} + \cdots + 2032282741 \)
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: |
\(913558883040682586951726894401\)
\(\medspace = 991^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(529.30\) | 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: | $991^{10/11}\approx 529.3017407164167$ | ||
| Ramified primes: |
\(991\)
| 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: | \(991\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{991}(1,·)$, $\chi_{991}(773,·)$, $\chi_{991}(134,·)$, $\chi_{991}(673,·)$, $\chi_{991}(42,·)$, $\chi_{991}(50,·)$, $\chi_{991}(945,·)$, $\chi_{991}(754,·)$, $\chi_{991}(518,·)$, $\chi_{991}(118,·)$, $\chi_{991}(947,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{113}a^{9}+\frac{4}{113}a^{8}-\frac{18}{113}a^{7}+\frac{30}{113}a^{6}-\frac{15}{113}a^{5}+\frac{9}{113}a^{4}+\frac{40}{113}a^{3}+\frac{13}{113}a^{2}+\frac{12}{113}a+\frac{17}{113}$, $\frac{1}{19\!\cdots\!23}a^{10}+\frac{16\!\cdots\!37}{19\!\cdots\!23}a^{9}-\frac{34\!\cdots\!10}{19\!\cdots\!23}a^{8}-\frac{33\!\cdots\!88}{19\!\cdots\!23}a^{7}+\frac{16\!\cdots\!84}{19\!\cdots\!23}a^{6}+\frac{71\!\cdots\!26}{19\!\cdots\!23}a^{5}+\frac{55\!\cdots\!09}{19\!\cdots\!23}a^{4}-\frac{46\!\cdots\!11}{19\!\cdots\!23}a^{3}+\frac{66\!\cdots\!39}{19\!\cdots\!23}a^{2}-\frac{63\!\cdots\!06}{19\!\cdots\!23}a-\frac{72\!\cdots\!61}{19\!\cdots\!23}$
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: |
$\frac{68\!\cdots\!18}{19\!\cdots\!23}a^{10}-\frac{13\!\cdots\!81}{19\!\cdots\!23}a^{9}-\frac{35\!\cdots\!15}{19\!\cdots\!23}a^{8}+\frac{64\!\cdots\!23}{19\!\cdots\!23}a^{7}+\frac{63\!\cdots\!69}{19\!\cdots\!23}a^{6}-\frac{99\!\cdots\!86}{19\!\cdots\!23}a^{5}-\frac{38\!\cdots\!38}{19\!\cdots\!23}a^{4}+\frac{58\!\cdots\!44}{19\!\cdots\!23}a^{3}+\frac{41\!\cdots\!43}{19\!\cdots\!23}a^{2}-\frac{11\!\cdots\!78}{19\!\cdots\!23}a+\frac{22\!\cdots\!65}{19\!\cdots\!23}$, $\frac{27\!\cdots\!95}{19\!\cdots\!23}a^{10}+\frac{92\!\cdots\!51}{19\!\cdots\!23}a^{9}-\frac{11\!\cdots\!28}{19\!\cdots\!23}a^{8}-\frac{42\!\cdots\!23}{19\!\cdots\!23}a^{7}+\frac{15\!\cdots\!58}{19\!\cdots\!23}a^{6}+\frac{52\!\cdots\!61}{19\!\cdots\!23}a^{5}-\frac{74\!\cdots\!80}{19\!\cdots\!23}a^{4}-\frac{17\!\cdots\!45}{19\!\cdots\!23}a^{3}+\frac{15\!\cdots\!74}{19\!\cdots\!23}a^{2}+\frac{17\!\cdots\!66}{19\!\cdots\!23}a-\frac{11\!\cdots\!63}{19\!\cdots\!23}$, $\frac{92\!\cdots\!61}{19\!\cdots\!23}a^{10}-\frac{64\!\cdots\!74}{19\!\cdots\!23}a^{9}-\frac{38\!\cdots\!40}{19\!\cdots\!23}a^{8}+\frac{26\!\cdots\!50}{19\!\cdots\!23}a^{7}+\frac{45\!\cdots\!74}{19\!\cdots\!23}a^{6}-\frac{33\!\cdots\!86}{19\!\cdots\!23}a^{5}-\frac{14\!\cdots\!65}{19\!\cdots\!23}a^{4}+\frac{14\!\cdots\!13}{19\!\cdots\!23}a^{3}-\frac{31\!\cdots\!58}{19\!\cdots\!23}a^{2}-\frac{16\!\cdots\!83}{19\!\cdots\!23}a+\frac{31\!\cdots\!50}{19\!\cdots\!23}$, $\frac{16\!\cdots\!95}{19\!\cdots\!23}a^{10}+\frac{63\!\cdots\!44}{19\!\cdots\!23}a^{9}-\frac{73\!\cdots\!74}{19\!\cdots\!23}a^{8}-\frac{28\!\cdots\!86}{19\!\cdots\!23}a^{7}+\frac{98\!\cdots\!43}{19\!\cdots\!23}a^{6}+\frac{33\!\cdots\!18}{19\!\cdots\!23}a^{5}-\frac{47\!\cdots\!35}{19\!\cdots\!23}a^{4}-\frac{11\!\cdots\!25}{19\!\cdots\!23}a^{3}+\frac{97\!\cdots\!69}{19\!\cdots\!23}a^{2}+\frac{11\!\cdots\!10}{19\!\cdots\!23}a-\frac{72\!\cdots\!08}{19\!\cdots\!23}$, $\frac{13\!\cdots\!93}{19\!\cdots\!23}a^{10}+\frac{52\!\cdots\!78}{19\!\cdots\!23}a^{9}-\frac{58\!\cdots\!89}{19\!\cdots\!23}a^{8}-\frac{23\!\cdots\!77}{19\!\cdots\!23}a^{7}+\frac{77\!\cdots\!13}{19\!\cdots\!23}a^{6}+\frac{27\!\cdots\!92}{19\!\cdots\!23}a^{5}-\frac{37\!\cdots\!51}{19\!\cdots\!23}a^{4}-\frac{92\!\cdots\!22}{19\!\cdots\!23}a^{3}+\frac{76\!\cdots\!87}{19\!\cdots\!23}a^{2}+\frac{91\!\cdots\!85}{19\!\cdots\!23}a-\frac{56\!\cdots\!55}{19\!\cdots\!23}$, $\frac{23\!\cdots\!70}{19\!\cdots\!23}a^{10}+\frac{84\!\cdots\!25}{19\!\cdots\!23}a^{9}-\frac{10\!\cdots\!27}{19\!\cdots\!23}a^{8}-\frac{37\!\cdots\!73}{19\!\cdots\!23}a^{7}+\frac{13\!\cdots\!69}{19\!\cdots\!23}a^{6}+\frac{44\!\cdots\!94}{19\!\cdots\!23}a^{5}-\frac{64\!\cdots\!10}{19\!\cdots\!23}a^{4}-\frac{14\!\cdots\!05}{19\!\cdots\!23}a^{3}+\frac{13\!\cdots\!60}{19\!\cdots\!23}a^{2}+\frac{13\!\cdots\!42}{19\!\cdots\!23}a-\frac{94\!\cdots\!48}{19\!\cdots\!23}$, $\frac{80\!\cdots\!25}{19\!\cdots\!23}a^{10}+\frac{30\!\cdots\!44}{19\!\cdots\!23}a^{9}-\frac{34\!\cdots\!60}{19\!\cdots\!23}a^{8}-\frac{13\!\cdots\!85}{19\!\cdots\!23}a^{7}+\frac{46\!\cdots\!27}{19\!\cdots\!23}a^{6}+\frac{16\!\cdots\!64}{19\!\cdots\!23}a^{5}-\frac{22\!\cdots\!14}{19\!\cdots\!23}a^{4}-\frac{54\!\cdots\!86}{19\!\cdots\!23}a^{3}+\frac{45\!\cdots\!39}{19\!\cdots\!23}a^{2}+\frac{54\!\cdots\!19}{19\!\cdots\!23}a-\frac{34\!\cdots\!52}{19\!\cdots\!23}$, $\frac{26\!\cdots\!21}{19\!\cdots\!23}a^{10}+\frac{10\!\cdots\!80}{19\!\cdots\!23}a^{9}-\frac{11\!\cdots\!70}{19\!\cdots\!23}a^{8}-\frac{47\!\cdots\!19}{19\!\cdots\!23}a^{7}+\frac{15\!\cdots\!53}{19\!\cdots\!23}a^{6}+\frac{57\!\cdots\!10}{19\!\cdots\!23}a^{5}-\frac{73\!\cdots\!73}{19\!\cdots\!23}a^{4}-\frac{19\!\cdots\!20}{19\!\cdots\!23}a^{3}+\frac{13\!\cdots\!10}{17\!\cdots\!71}a^{2}+\frac{18\!\cdots\!06}{19\!\cdots\!23}a-\frac{11\!\cdots\!32}{19\!\cdots\!23}$, $\frac{22\!\cdots\!24}{19\!\cdots\!23}a^{10}-\frac{29\!\cdots\!13}{19\!\cdots\!23}a^{9}-\frac{97\!\cdots\!45}{19\!\cdots\!23}a^{8}+\frac{11\!\cdots\!00}{19\!\cdots\!23}a^{7}+\frac{12\!\cdots\!92}{19\!\cdots\!23}a^{6}-\frac{19\!\cdots\!14}{19\!\cdots\!23}a^{5}-\frac{55\!\cdots\!83}{19\!\cdots\!23}a^{4}+\frac{13\!\cdots\!47}{19\!\cdots\!23}a^{3}+\frac{74\!\cdots\!23}{19\!\cdots\!23}a^{2}-\frac{24\!\cdots\!41}{19\!\cdots\!23}a+\frac{99\!\cdots\!42}{19\!\cdots\!23}$, $\frac{11\!\cdots\!58}{19\!\cdots\!23}a^{10}-\frac{77\!\cdots\!24}{19\!\cdots\!23}a^{9}-\frac{45\!\cdots\!80}{19\!\cdots\!23}a^{8}+\frac{30\!\cdots\!65}{19\!\cdots\!23}a^{7}+\frac{53\!\cdots\!80}{19\!\cdots\!23}a^{6}-\frac{37\!\cdots\!41}{19\!\cdots\!23}a^{5}-\frac{15\!\cdots\!21}{19\!\cdots\!23}a^{4}+\frac{14\!\cdots\!69}{19\!\cdots\!23}a^{3}+\frac{28\!\cdots\!75}{19\!\cdots\!23}a^{2}-\frac{15\!\cdots\!82}{19\!\cdots\!23}a+\frac{24\!\cdots\!87}{19\!\cdots\!23}$
| 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: | \( 155801246971 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 155801246971 \cdot 1}{2\cdot\sqrt{913558883040682586951726894401}}\cr\approx \mathstrut & 0.166917785190651 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 450*x^9 + 393*x^8 + 65785*x^7 - 76121*x^6 - 3728249*x^5 + 6483322*x^4 + 89323866*x^3 - 205338716*x^2 - 745717454*x + 2032282741)
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 - 450*x^9 + 393*x^8 + 65785*x^7 - 76121*x^6 - 3728249*x^5 + 6483322*x^4 + 89323866*x^3 - 205338716*x^2 - 745717454*x + 2032282741, 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 - 450*x^9 + 393*x^8 + 65785*x^7 - 76121*x^6 - 3728249*x^5 + 6483322*x^4 + 89323866*x^3 - 205338716*x^2 - 745717454*x + 2032282741);
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 - 450*x^9 + 393*x^8 + 65785*x^7 - 76121*x^6 - 3728249*x^5 + 6483322*x^4 + 89323866*x^3 - 205338716*x^2 - 745717454*x + 2032282741);
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 | ${\href{/padicField/2.11.0.1}{11} }$ | ${\href{/padicField/3.11.0.1}{11} }$ | ${\href{/padicField/5.11.0.1}{11} }$ | ${\href{/padicField/7.11.0.1}{11} }$ | ${\href{/padicField/11.11.0.1}{11} }$ | ${\href{/padicField/13.11.0.1}{11} }$ | ${\href{/padicField/17.11.0.1}{11} }$ | ${\href{/padicField/19.11.0.1}{11} }$ | ${\href{/padicField/23.11.0.1}{11} }$ | ${\href{/padicField/29.11.0.1}{11} }$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.11.0.1}{11} }$ | ${\href{/padicField/41.11.0.1}{11} }$ | ${\href{/padicField/43.11.0.1}{11} }$ | ${\href{/padicField/47.11.0.1}{11} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.1.0.1}{1} }^{11}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(991\)
| Deg $11$ | $11$ | $1$ | $10$ |