Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 310*x^9 + 395*x^8 + 26441*x^7 - 57583*x^6 - 744575*x^5 + 2220564*x^4 + 2568726*x^3 - 5088204*x^2 - 1360638*x + 996543)
gp: K = bnfinit(y^11 - y^10 - 310*y^9 + 395*y^8 + 26441*y^7 - 57583*y^6 - 744575*y^5 + 2220564*y^4 + 2568726*y^3 - 5088204*y^2 - 1360638*y + 996543, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - x^10 - 310*x^9 + 395*x^8 + 26441*x^7 - 57583*x^6 - 744575*x^5 + 2220564*x^4 + 2568726*x^3 - 5088204*x^2 - 1360638*x + 996543);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^11 - x^10 - 310*x^9 + 395*x^8 + 26441*x^7 - 57583*x^6 - 744575*x^5 + 2220564*x^4 + 2568726*x^3 - 5088204*x^2 - 1360638*x + 996543)
\( x^{11} - x^{10} - 310 x^{9} + 395 x^{8} + 26441 x^{7} - 57583 x^{6} - 744575 x^{5} + 2220564 x^{4} + \cdots + 996543 \)
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: |
\(22090575837180674640752471449\)
\(\medspace = 683^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(377.35\) | 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: | $683^{10/11}\approx 377.351511792966$ | ||
| Ramified primes: |
\(683\)
| 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: | \(683\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{683}(64,·)$, $\chi_{683}(1,·)$, $\chi_{683}(555,·)$, $\chi_{683}(4,·)$, $\chi_{683}(681,·)$, $\chi_{683}(651,·)$, $\chi_{683}(256,·)$, $\chi_{683}(16,·)$, $\chi_{683}(675,·)$, $\chi_{683}(171,·)$, $\chi_{683}(341,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{5}-\frac{1}{3}a$, $\frac{1}{9}a^{6}+\frac{1}{9}a^{4}-\frac{2}{9}a^{2}$, $\frac{1}{27}a^{7}+\frac{1}{27}a^{5}+\frac{1}{9}a^{4}-\frac{2}{27}a^{3}-\frac{4}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{81}a^{8}+\frac{1}{81}a^{7}+\frac{4}{81}a^{6}+\frac{4}{81}a^{5}+\frac{13}{81}a^{4}+\frac{4}{81}a^{3}+\frac{1}{9}a^{2}-\frac{4}{9}a$, $\frac{1}{2997}a^{9}-\frac{5}{999}a^{8}-\frac{14}{999}a^{7}+\frac{13}{999}a^{6}+\frac{11}{111}a^{5}-\frac{56}{999}a^{4}-\frac{130}{2997}a^{3}-\frac{86}{333}a^{2}-\frac{164}{333}a-\frac{15}{37}$, $\frac{1}{30\!\cdots\!03}a^{10}+\frac{65\!\cdots\!83}{30\!\cdots\!03}a^{9}-\frac{14\!\cdots\!05}{30\!\cdots\!03}a^{8}+\frac{38\!\cdots\!98}{30\!\cdots\!03}a^{7}-\frac{16\!\cdots\!30}{30\!\cdots\!03}a^{6}-\frac{18\!\cdots\!38}{30\!\cdots\!03}a^{5}-\frac{30\!\cdots\!00}{30\!\cdots\!03}a^{4}-\frac{13\!\cdots\!96}{10\!\cdots\!01}a^{3}-\frac{16\!\cdots\!90}{34\!\cdots\!67}a^{2}+\frac{91\!\cdots\!52}{11\!\cdots\!89}a-\frac{55\!\cdots\!77}{12\!\cdots\!21}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
$\frac{15\!\cdots\!45}{15\!\cdots\!03}a^{10}-\frac{33\!\cdots\!89}{15\!\cdots\!03}a^{9}-\frac{44\!\cdots\!44}{15\!\cdots\!03}a^{8}+\frac{90\!\cdots\!16}{15\!\cdots\!03}a^{7}+\frac{32\!\cdots\!49}{15\!\cdots\!03}a^{6}-\frac{70\!\cdots\!25}{15\!\cdots\!03}a^{5}-\frac{78\!\cdots\!05}{15\!\cdots\!03}a^{4}+\frac{49\!\cdots\!05}{51\!\cdots\!01}a^{3}+\frac{49\!\cdots\!48}{17\!\cdots\!67}a^{2}+\frac{15\!\cdots\!58}{57\!\cdots\!89}a-\frac{32\!\cdots\!51}{63\!\cdots\!21}$, $\frac{10\!\cdots\!39}{30\!\cdots\!03}a^{10}+\frac{46\!\cdots\!19}{30\!\cdots\!03}a^{9}-\frac{28\!\cdots\!48}{30\!\cdots\!03}a^{8}-\frac{12\!\cdots\!79}{30\!\cdots\!03}a^{7}+\frac{20\!\cdots\!75}{30\!\cdots\!03}a^{6}+\frac{57\!\cdots\!24}{30\!\cdots\!03}a^{5}-\frac{46\!\cdots\!34}{30\!\cdots\!03}a^{4}-\frac{15\!\cdots\!40}{10\!\cdots\!01}a^{3}+\frac{97\!\cdots\!72}{34\!\cdots\!67}a^{2}+\frac{18\!\cdots\!50}{11\!\cdots\!89}a-\frac{63\!\cdots\!61}{12\!\cdots\!21}$, $\frac{57\!\cdots\!39}{10\!\cdots\!01}a^{10}+\frac{23\!\cdots\!90}{10\!\cdots\!01}a^{9}-\frac{16\!\cdots\!63}{10\!\cdots\!01}a^{8}-\frac{61\!\cdots\!87}{10\!\cdots\!01}a^{7}+\frac{11\!\cdots\!17}{10\!\cdots\!01}a^{6}+\frac{27\!\cdots\!68}{10\!\cdots\!01}a^{5}-\frac{25\!\cdots\!37}{10\!\cdots\!01}a^{4}-\frac{52\!\cdots\!32}{34\!\cdots\!67}a^{3}+\frac{59\!\cdots\!13}{11\!\cdots\!89}a^{2}+\frac{43\!\cdots\!07}{38\!\cdots\!63}a-\frac{41\!\cdots\!95}{42\!\cdots\!07}$, $\frac{78\!\cdots\!97}{30\!\cdots\!03}a^{10}+\frac{68\!\cdots\!98}{30\!\cdots\!03}a^{9}-\frac{20\!\cdots\!86}{30\!\cdots\!03}a^{8}-\frac{17\!\cdots\!52}{30\!\cdots\!03}a^{7}+\frac{10\!\cdots\!24}{30\!\cdots\!03}a^{6}+\frac{89\!\cdots\!74}{30\!\cdots\!03}a^{5}-\frac{14\!\cdots\!85}{30\!\cdots\!03}a^{4}-\frac{40\!\cdots\!77}{10\!\cdots\!01}a^{3}-\frac{56\!\cdots\!50}{34\!\cdots\!67}a^{2}+\frac{76\!\cdots\!33}{11\!\cdots\!89}a+\frac{45\!\cdots\!06}{12\!\cdots\!21}$, $\frac{93\!\cdots\!25}{30\!\cdots\!03}a^{10}-\frac{19\!\cdots\!31}{30\!\cdots\!03}a^{9}-\frac{28\!\cdots\!78}{30\!\cdots\!03}a^{8}+\frac{70\!\cdots\!77}{30\!\cdots\!03}a^{7}+\frac{23\!\cdots\!80}{30\!\cdots\!03}a^{6}-\frac{85\!\cdots\!55}{30\!\cdots\!03}a^{5}-\frac{58\!\cdots\!28}{30\!\cdots\!03}a^{4}+\frac{98\!\cdots\!31}{10\!\cdots\!01}a^{3}-\frac{24\!\cdots\!14}{34\!\cdots\!67}a^{2}-\frac{43\!\cdots\!22}{11\!\cdots\!89}a+\frac{24\!\cdots\!19}{12\!\cdots\!21}$, $\frac{14\!\cdots\!18}{34\!\cdots\!67}a^{10}-\frac{23\!\cdots\!24}{34\!\cdots\!67}a^{9}-\frac{45\!\cdots\!48}{34\!\cdots\!67}a^{8}+\frac{86\!\cdots\!93}{34\!\cdots\!67}a^{7}+\frac{39\!\cdots\!67}{34\!\cdots\!67}a^{6}-\frac{10\!\cdots\!98}{34\!\cdots\!67}a^{5}-\frac{10\!\cdots\!77}{34\!\cdots\!67}a^{4}+\frac{12\!\cdots\!48}{11\!\cdots\!89}a^{3}+\frac{23\!\cdots\!24}{38\!\cdots\!63}a^{2}-\frac{29\!\cdots\!56}{12\!\cdots\!21}a+\frac{30\!\cdots\!29}{42\!\cdots\!07}$, $\frac{18\!\cdots\!09}{30\!\cdots\!03}a^{10}+\frac{10\!\cdots\!74}{30\!\cdots\!03}a^{9}-\frac{65\!\cdots\!41}{30\!\cdots\!03}a^{8}-\frac{24\!\cdots\!13}{30\!\cdots\!03}a^{7}+\frac{71\!\cdots\!53}{30\!\cdots\!03}a^{6}+\frac{67\!\cdots\!68}{30\!\cdots\!03}a^{5}-\frac{26\!\cdots\!58}{30\!\cdots\!03}a^{4}+\frac{11\!\cdots\!77}{10\!\cdots\!01}a^{3}+\frac{15\!\cdots\!60}{34\!\cdots\!67}a^{2}+\frac{72\!\cdots\!14}{11\!\cdots\!89}a-\frac{98\!\cdots\!26}{12\!\cdots\!21}$, $\frac{10\!\cdots\!53}{30\!\cdots\!03}a^{10}+\frac{50\!\cdots\!82}{30\!\cdots\!03}a^{9}-\frac{30\!\cdots\!73}{30\!\cdots\!03}a^{8}-\frac{13\!\cdots\!33}{30\!\cdots\!03}a^{7}+\frac{21\!\cdots\!54}{30\!\cdots\!03}a^{6}+\frac{60\!\cdots\!36}{30\!\cdots\!03}a^{5}-\frac{48\!\cdots\!08}{30\!\cdots\!03}a^{4}-\frac{13\!\cdots\!40}{10\!\cdots\!01}a^{3}+\frac{11\!\cdots\!27}{34\!\cdots\!67}a^{2}+\frac{16\!\cdots\!90}{11\!\cdots\!89}a-\frac{11\!\cdots\!25}{12\!\cdots\!21}$, $\frac{50\!\cdots\!29}{30\!\cdots\!03}a^{10}+\frac{30\!\cdots\!36}{30\!\cdots\!03}a^{9}-\frac{13\!\cdots\!52}{30\!\cdots\!03}a^{8}-\frac{75\!\cdots\!65}{30\!\cdots\!03}a^{7}+\frac{88\!\cdots\!15}{30\!\cdots\!03}a^{6}+\frac{30\!\cdots\!34}{30\!\cdots\!03}a^{5}-\frac{19\!\cdots\!45}{30\!\cdots\!03}a^{4}-\frac{12\!\cdots\!81}{27\!\cdots\!73}a^{3}+\frac{47\!\cdots\!36}{34\!\cdots\!67}a^{2}+\frac{36\!\cdots\!66}{11\!\cdots\!89}a-\frac{33\!\cdots\!54}{12\!\cdots\!21}$, $\frac{47\!\cdots\!55}{10\!\cdots\!01}a^{10}+\frac{26\!\cdots\!13}{10\!\cdots\!01}a^{9}-\frac{12\!\cdots\!46}{10\!\cdots\!01}a^{8}-\frac{65\!\cdots\!13}{10\!\cdots\!01}a^{7}+\frac{81\!\cdots\!40}{10\!\cdots\!01}a^{6}+\frac{26\!\cdots\!84}{10\!\cdots\!01}a^{5}-\frac{17\!\cdots\!81}{10\!\cdots\!01}a^{4}-\frac{40\!\cdots\!14}{34\!\cdots\!67}a^{3}+\frac{41\!\cdots\!46}{11\!\cdots\!89}a^{2}+\frac{31\!\cdots\!86}{38\!\cdots\!63}a-\frac{29\!\cdots\!14}{42\!\cdots\!07}$
| 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: | \( 2276941413140 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 2276941413140 \cdot 1}{2\cdot\sqrt{22090575837180674640752471449}}\cr\approx \mathstrut & 15.6873033378122 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 310*x^9 + 395*x^8 + 26441*x^7 - 57583*x^6 - 744575*x^5 + 2220564*x^4 + 2568726*x^3 - 5088204*x^2 - 1360638*x + 996543)
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 - 310*x^9 + 395*x^8 + 26441*x^7 - 57583*x^6 - 744575*x^5 + 2220564*x^4 + 2568726*x^3 - 5088204*x^2 - 1360638*x + 996543, 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 - 310*x^9 + 395*x^8 + 26441*x^7 - 57583*x^6 - 744575*x^5 + 2220564*x^4 + 2568726*x^3 - 5088204*x^2 - 1360638*x + 996543);
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 - 310*x^9 + 395*x^8 + 26441*x^7 - 57583*x^6 - 744575*x^5 + 2220564*x^4 + 2568726*x^3 - 5088204*x^2 - 1360638*x + 996543);
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 | ${\href{/padicField/2.11.0.1}{11} }$ | ${\href{/padicField/3.1.0.1}{1} }^{11}$ | ${\href{/padicField/5.11.0.1}{11} }$ | ${\href{/padicField/7.11.0.1}{11} }$ | ${\href{/padicField/11.11.0.1}{11} }$ | ${\href{/padicField/13.11.0.1}{11} }$ | ${\href{/padicField/17.11.0.1}{11} }$ | ${\href{/padicField/19.11.0.1}{11} }$ | ${\href{/padicField/23.11.0.1}{11} }$ | ${\href{/padicField/29.11.0.1}{11} }$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.1.0.1}{1} }^{11}$ | ${\href{/padicField/41.11.0.1}{11} }$ | ${\href{/padicField/43.11.0.1}{11} }$ | ${\href{/padicField/47.11.0.1}{11} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.11.0.1}{11} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(683\)
| Deg $11$ | $11$ | $1$ | $10$ |