Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 110*x^9 - 484*x^8 + 5401*x^7 + 22484*x^6 - 116248*x^5 - 453904*x^4 - 2387968*x^3 - 6994944*x^2 - 7118848*x - 2506752)
gp: K = bnfinit(y^11 - 110*y^9 - 484*y^8 + 5401*y^7 + 22484*y^6 - 116248*y^5 - 453904*y^4 - 2387968*y^3 - 6994944*y^2 - 7118848*y - 2506752, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 110*x^9 - 484*x^8 + 5401*x^7 + 22484*x^6 - 116248*x^5 - 453904*x^4 - 2387968*x^3 - 6994944*x^2 - 7118848*x - 2506752);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^11 - 110*x^9 - 484*x^8 + 5401*x^7 + 22484*x^6 - 116248*x^5 - 453904*x^4 - 2387968*x^3 - 6994944*x^2 - 7118848*x - 2506752)
\( x^{11} - 110 x^{9} - 484 x^{8} + 5401 x^{7} + 22484 x^{6} - 116248 x^{5} - 453904 x^{4} - 2387968 x^{3} + \cdots - 2506752 \)
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: | $[3, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(5057930951035035239722254336\)
\(\medspace = 2^{24}\cdot 3^{8}\cdot 11^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(330.02\) | 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: | $2^{19/6}3^{7/6}11^{84/55}\approx 1260.067363427052$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{16}a^{6}-\frac{1}{16}a^{5}+\frac{1}{16}a^{4}+\frac{3}{16}a^{3}$, $\frac{1}{16}a^{7}+\frac{3}{16}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{8}-\frac{1}{32}a^{6}+\frac{5}{64}a^{4}+\frac{1}{8}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{1024}a^{9}-\frac{7}{512}a^{7}+\frac{7}{256}a^{6}-\frac{39}{1024}a^{5}+\frac{21}{256}a^{4}-\frac{23}{128}a^{3}-\frac{25}{64}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{83\!\cdots\!40}a^{10}-\frac{17\!\cdots\!19}{10\!\cdots\!80}a^{9}+\frac{51\!\cdots\!37}{41\!\cdots\!20}a^{8}-\frac{25\!\cdots\!91}{69\!\cdots\!20}a^{7}+\frac{19\!\cdots\!49}{16\!\cdots\!08}a^{6}+\frac{54\!\cdots\!31}{20\!\cdots\!60}a^{5}-\frac{21\!\cdots\!87}{10\!\cdots\!80}a^{4}+\frac{48\!\cdots\!03}{52\!\cdots\!40}a^{3}+\frac{91\!\cdots\!37}{65\!\cdots\!80}a^{2}+\frac{87\!\cdots\!17}{16\!\cdots\!20}a-\frac{23\!\cdots\!87}{68\!\cdots\!30}$
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
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: | $6$ | 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: |
$\frac{26\!\cdots\!53}{83\!\cdots\!40}a^{10}-\frac{81\!\cdots\!29}{13\!\cdots\!60}a^{9}-\frac{14\!\cdots\!59}{41\!\cdots\!20}a^{8}-\frac{61\!\cdots\!03}{69\!\cdots\!20}a^{7}+\frac{31\!\cdots\!09}{16\!\cdots\!08}a^{6}+\frac{71\!\cdots\!73}{20\!\cdots\!60}a^{5}-\frac{45\!\cdots\!91}{10\!\cdots\!80}a^{4}-\frac{29\!\cdots\!81}{52\!\cdots\!40}a^{3}-\frac{21\!\cdots\!47}{32\!\cdots\!40}a^{2}-\frac{15\!\cdots\!59}{16\!\cdots\!20}a-\frac{13\!\cdots\!63}{34\!\cdots\!65}$, $\frac{55\!\cdots\!47}{13\!\cdots\!40}a^{10}+\frac{48\!\cdots\!97}{17\!\cdots\!80}a^{9}-\frac{32\!\cdots\!01}{69\!\cdots\!20}a^{8}-\frac{18\!\cdots\!71}{34\!\cdots\!60}a^{7}+\frac{34\!\cdots\!15}{27\!\cdots\!68}a^{6}+\frac{10\!\cdots\!77}{34\!\cdots\!60}a^{5}+\frac{10\!\cdots\!71}{17\!\cdots\!80}a^{4}-\frac{67\!\cdots\!99}{87\!\cdots\!40}a^{3}-\frac{25\!\cdots\!91}{10\!\cdots\!80}a^{2}-\frac{60\!\cdots\!01}{27\!\cdots\!20}a-\frac{25\!\cdots\!22}{34\!\cdots\!65}$, $\frac{17\!\cdots\!53}{41\!\cdots\!52}a^{10}+\frac{86\!\cdots\!49}{20\!\cdots\!76}a^{9}-\frac{74\!\cdots\!91}{20\!\cdots\!76}a^{8}-\frac{10\!\cdots\!97}{43\!\cdots\!12}a^{7}-\frac{56\!\cdots\!27}{41\!\cdots\!52}a^{6}+\frac{16\!\cdots\!87}{20\!\cdots\!76}a^{5}+\frac{81\!\cdots\!03}{26\!\cdots\!72}a^{4}+\frac{20\!\cdots\!41}{16\!\cdots\!92}a^{3}+\frac{40\!\cdots\!47}{13\!\cdots\!36}a^{2}+\frac{60\!\cdots\!94}{20\!\cdots\!99}a+\frac{13\!\cdots\!85}{13\!\cdots\!66}$, $\frac{58\!\cdots\!87}{41\!\cdots\!20}a^{10}-\frac{11\!\cdots\!61}{10\!\cdots\!80}a^{9}-\frac{21\!\cdots\!41}{20\!\cdots\!60}a^{8}+\frac{24\!\cdots\!73}{34\!\cdots\!60}a^{7}+\frac{19\!\cdots\!99}{83\!\cdots\!04}a^{6}-\frac{53\!\cdots\!19}{52\!\cdots\!40}a^{5}-\frac{20\!\cdots\!59}{52\!\cdots\!40}a^{4}-\frac{62\!\cdots\!49}{26\!\cdots\!20}a^{3}-\frac{46\!\cdots\!47}{65\!\cdots\!80}a^{2}-\frac{58\!\cdots\!61}{81\!\cdots\!60}a-\frac{17\!\cdots\!93}{68\!\cdots\!30}$, $\frac{43\!\cdots\!13}{41\!\cdots\!20}a^{10}-\frac{21\!\cdots\!09}{10\!\cdots\!80}a^{9}-\frac{23\!\cdots\!99}{20\!\cdots\!60}a^{8}-\frac{99\!\cdots\!33}{34\!\cdots\!60}a^{7}+\frac{52\!\cdots\!65}{83\!\cdots\!04}a^{6}+\frac{11\!\cdots\!97}{10\!\cdots\!95}a^{5}-\frac{75\!\cdots\!61}{52\!\cdots\!40}a^{4}-\frac{49\!\cdots\!11}{26\!\cdots\!20}a^{3}-\frac{13\!\cdots\!43}{65\!\cdots\!80}a^{2}-\frac{25\!\cdots\!89}{81\!\cdots\!60}a-\frac{89\!\cdots\!17}{68\!\cdots\!30}$, $\frac{67\!\cdots\!83}{41\!\cdots\!52}a^{10}-\frac{33\!\cdots\!17}{10\!\cdots\!88}a^{9}-\frac{35\!\cdots\!33}{20\!\cdots\!76}a^{8}-\frac{15\!\cdots\!39}{34\!\cdots\!96}a^{7}+\frac{40\!\cdots\!71}{41\!\cdots\!52}a^{6}+\frac{89\!\cdots\!47}{52\!\cdots\!44}a^{5}-\frac{11\!\cdots\!23}{52\!\cdots\!44}a^{4}-\frac{75\!\cdots\!77}{26\!\cdots\!72}a^{3}-\frac{21\!\cdots\!71}{65\!\cdots\!68}a^{2}-\frac{96\!\cdots\!02}{20\!\cdots\!99}a-\frac{13\!\cdots\!70}{68\!\cdots\!33}$
| 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: | \( 115991956666 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 115991956666 \cdot 1}{2\cdot\sqrt{5057930951035035239722254336}}\cr\approx \mathstrut & 10.1676570966307 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^11 - 110*x^9 - 484*x^8 + 5401*x^7 + 22484*x^6 - 116248*x^5 - 453904*x^4 - 2387968*x^3 - 6994944*x^2 - 7118848*x - 2506752)
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 - 110*x^9 - 484*x^8 + 5401*x^7 + 22484*x^6 - 116248*x^5 - 453904*x^4 - 2387968*x^3 - 6994944*x^2 - 7118848*x - 2506752, 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 - 110*x^9 - 484*x^8 + 5401*x^7 + 22484*x^6 - 116248*x^5 - 453904*x^4 - 2387968*x^3 - 6994944*x^2 - 7118848*x - 2506752);
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 - 110*x^9 - 484*x^8 + 5401*x^7 + 22484*x^6 - 116248*x^5 - 453904*x^4 - 2387968*x^3 - 6994944*x^2 - 7118848*x - 2506752);
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 non-solvable group of order 19958400 |
| The 31 conjugacy class representatives for $A_{11}$ |
| Character table for $A_{11}$ |
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 | R | R | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }$ | ${\href{/padicField/29.11.0.1}{11} }$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.11.0.1}{11} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.4.11.12 | $x^{4} + 26$ | $4$ | $1$ | $11$ | $D_{4}$ | $[2, 3, 4]$ | |
| 2.4.11.5 | $x^{4} + 2$ | $4$ | $1$ | $11$ | $D_{4}$ | $[2, 3, 4]$ | |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.6.6.3 | $x^{6} + 18 x^{5} + 120 x^{4} + 386 x^{3} + 723 x^{2} + 732 x + 305$ | $3$ | $2$ | $6$ | $D_{6}$ | $[3/2]_{2}^{2}$ | |
|
\(11\)
| 11.11.16.4 | $x^{11} + 22 x^{6} + 11$ | $11$ | $1$ | $16$ | $C_{11}:C_5$ | $[8/5]_{5}$ |