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Properties

Label 11.11.580...049.1
Degree $11$
Signature $[11, 0]$
Discriminant $5.801\times 10^{29}$
Root discriminant \(507.89\)
Ramified prime $947$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{11}$ (as 11T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 430*x^9 - 2035*x^8 + 28780*x^7 + 141142*x^6 - 760875*x^5 - 3028650*x^4 + 8674010*x^3 + 20981317*x^2 - 30077229*x - 47440673)
 
gp: K = bnfinit(y^11 - y^10 - 430*y^9 - 2035*y^8 + 28780*y^7 + 141142*y^6 - 760875*y^5 - 3028650*y^4 + 8674010*y^3 + 20981317*y^2 - 30077229*y - 47440673, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - x^10 - 430*x^9 - 2035*x^8 + 28780*x^7 + 141142*x^6 - 760875*x^5 - 3028650*x^4 + 8674010*x^3 + 20981317*x^2 - 30077229*x - 47440673);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^11 - x^10 - 430*x^9 - 2035*x^8 + 28780*x^7 + 141142*x^6 - 760875*x^5 - 3028650*x^4 + 8674010*x^3 + 20981317*x^2 - 30077229*x - 47440673)
 

\( x^{11} - x^{10} - 430 x^{9} - 2035 x^{8} + 28780 x^{7} + 141142 x^{6} - 760875 x^{5} - 3028650 x^{4} + \cdots - 47440673 \) Copy content Toggle raw display

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:   \(580095892065127629623589183049\) \(\medspace = 947^{10}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(507.89\)
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:  $947^{10/11}\approx 507.8935624536204$
Ramified primes:   \(947\) Copy content Toggle raw display
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:  \(947\)
Dirichlet character group:    $\lbrace$$\chi_{947}(1,·)$, $\chi_{947}(930,·)$, $\chi_{947}(643,·)$, $\chi_{947}(580,·)$, $\chi_{947}(133,·)$, $\chi_{947}(289,·)$, $\chi_{947}(557,·)$, $\chi_{947}(433,·)$, $\chi_{947}(215,·)$, $\chi_{947}(185,·)$, $\chi_{947}(769,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{2}{7}a^{5}-\frac{3}{7}a^{4}-\frac{1}{7}a^{3}+\frac{2}{7}a^{2}+\frac{3}{7}a$, $\frac{1}{7}a^{7}-\frac{1}{7}a$, $\frac{1}{2303}a^{8}+\frac{32}{2303}a^{7}-\frac{64}{2303}a^{6}-\frac{1048}{2303}a^{5}-\frac{1117}{2303}a^{4}-\frac{1035}{2303}a^{3}-\frac{1088}{2303}a^{2}+\frac{36}{329}a+\frac{9}{47}$, $\frac{1}{12972799}a^{9}+\frac{955}{12972799}a^{8}+\frac{14205}{301693}a^{7}-\frac{307199}{12972799}a^{6}+\frac{359423}{12972799}a^{5}+\frac{102789}{276017}a^{4}+\frac{4732675}{12972799}a^{3}+\frac{2642664}{12972799}a^{2}+\frac{778382}{1853257}a+\frac{127358}{264751}$, $\frac{1}{14\!\cdots\!19}a^{10}-\frac{483063088171}{14\!\cdots\!19}a^{9}-\frac{25\!\cdots\!42}{14\!\cdots\!19}a^{8}-\frac{37\!\cdots\!63}{14\!\cdots\!19}a^{7}+\frac{75\!\cdots\!47}{14\!\cdots\!19}a^{6}-\frac{41\!\cdots\!92}{14\!\cdots\!19}a^{5}-\frac{55\!\cdots\!93}{29\!\cdots\!31}a^{4}-\frac{31\!\cdots\!04}{14\!\cdots\!19}a^{3}-\frac{38\!\cdots\!78}{14\!\cdots\!19}a^{2}-\frac{57\!\cdots\!70}{20\!\cdots\!17}a+\frac{13\!\cdots\!82}{29\!\cdots\!31}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $7$

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 $2$) Copy content Toggle raw display
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:   $\frac{214617842505333}{14\!\cdots\!19}a^{10}+\frac{628159799258032}{14\!\cdots\!19}a^{9}-\frac{91\!\cdots\!28}{14\!\cdots\!19}a^{8}-\frac{79\!\cdots\!15}{14\!\cdots\!19}a^{7}+\frac{36\!\cdots\!32}{14\!\cdots\!19}a^{6}+\frac{48\!\cdots\!71}{14\!\cdots\!19}a^{5}-\frac{53\!\cdots\!82}{14\!\cdots\!19}a^{4}-\frac{13\!\cdots\!44}{20\!\cdots\!17}a^{3}-\frac{11\!\cdots\!71}{14\!\cdots\!19}a^{2}+\frac{73\!\cdots\!19}{20\!\cdots\!17}a+\frac{15\!\cdots\!73}{29\!\cdots\!31}$, $\frac{21540014079238}{20\!\cdots\!17}a^{10}+\frac{5820452854204}{20\!\cdots\!17}a^{9}-\frac{62\!\cdots\!83}{14\!\cdots\!19}a^{8}-\frac{39\!\cdots\!44}{14\!\cdots\!19}a^{7}+\frac{22\!\cdots\!13}{10\!\cdots\!49}a^{6}+\frac{23\!\cdots\!98}{14\!\cdots\!19}a^{5}-\frac{29\!\cdots\!69}{14\!\cdots\!19}a^{4}-\frac{41\!\cdots\!87}{14\!\cdots\!19}a^{3}-\frac{41\!\cdots\!40}{14\!\cdots\!19}a^{2}+\frac{26\!\cdots\!67}{20\!\cdots\!17}a+\frac{66\!\cdots\!12}{29\!\cdots\!31}$, $\frac{16285808980254}{14\!\cdots\!19}a^{10}-\frac{202284819340265}{14\!\cdots\!19}a^{9}-\frac{68\!\cdots\!43}{14\!\cdots\!19}a^{8}+\frac{45\!\cdots\!36}{14\!\cdots\!19}a^{7}+\frac{85\!\cdots\!49}{14\!\cdots\!19}a^{6}-\frac{25\!\cdots\!67}{14\!\cdots\!19}a^{5}-\frac{36\!\cdots\!16}{14\!\cdots\!19}a^{4}+\frac{62\!\cdots\!26}{14\!\cdots\!19}a^{3}+\frac{77\!\cdots\!92}{20\!\cdots\!17}a^{2}-\frac{91\!\cdots\!71}{20\!\cdots\!17}a-\frac{29\!\cdots\!19}{29\!\cdots\!31}$, $\frac{6225644862783}{14\!\cdots\!19}a^{10}+\frac{215043742063361}{14\!\cdots\!19}a^{9}+\frac{22\!\cdots\!88}{14\!\cdots\!19}a^{8}+\frac{231274557235003}{20\!\cdots\!17}a^{7}-\frac{14\!\cdots\!60}{20\!\cdots\!17}a^{6}-\frac{61\!\cdots\!27}{20\!\cdots\!17}a^{5}+\frac{10\!\cdots\!40}{14\!\cdots\!19}a^{4}+\frac{75\!\cdots\!13}{14\!\cdots\!19}a^{3}+\frac{22\!\cdots\!35}{14\!\cdots\!19}a^{2}-\frac{48\!\cdots\!47}{20\!\cdots\!17}a-\frac{70\!\cdots\!54}{29\!\cdots\!31}$, $\frac{53553813041487}{14\!\cdots\!19}a^{10}-\frac{486100484182958}{14\!\cdots\!19}a^{9}-\frac{28\!\cdots\!07}{20\!\cdots\!17}a^{8}+\frac{52\!\cdots\!98}{14\!\cdots\!19}a^{7}+\frac{13\!\cdots\!12}{14\!\cdots\!19}a^{6}-\frac{28\!\cdots\!47}{14\!\cdots\!19}a^{5}-\frac{30\!\cdots\!01}{14\!\cdots\!19}a^{4}+\frac{60\!\cdots\!27}{14\!\cdots\!19}a^{3}+\frac{20\!\cdots\!50}{14\!\cdots\!19}a^{2}-\frac{39\!\cdots\!48}{20\!\cdots\!17}a-\frac{96\!\cdots\!97}{29\!\cdots\!31}$, $\frac{250375949956522}{14\!\cdots\!19}a^{10}+\frac{290701362023623}{14\!\cdots\!19}a^{9}-\frac{10\!\cdots\!19}{14\!\cdots\!19}a^{8}-\frac{74\!\cdots\!69}{14\!\cdots\!19}a^{7}+\frac{51\!\cdots\!21}{14\!\cdots\!19}a^{6}+\frac{46\!\cdots\!96}{14\!\cdots\!19}a^{5}-\frac{56\!\cdots\!12}{14\!\cdots\!19}a^{4}-\frac{86\!\cdots\!87}{14\!\cdots\!19}a^{3}-\frac{38\!\cdots\!66}{14\!\cdots\!19}a^{2}+\frac{12\!\cdots\!45}{47\!\cdots\!19}a+\frac{78\!\cdots\!95}{29\!\cdots\!31}$, $\frac{158361460978612}{14\!\cdots\!19}a^{10}+\frac{1767592296620}{29\!\cdots\!31}a^{9}-\frac{220826463283560}{47\!\cdots\!19}a^{8}-\frac{41\!\cdots\!02}{14\!\cdots\!19}a^{7}+\frac{32\!\cdots\!34}{14\!\cdots\!19}a^{6}+\frac{21\!\cdots\!11}{14\!\cdots\!19}a^{5}-\frac{69\!\cdots\!17}{14\!\cdots\!19}a^{4}-\frac{33\!\cdots\!17}{14\!\cdots\!19}a^{3}+\frac{82\!\cdots\!06}{14\!\cdots\!19}a^{2}+\frac{17\!\cdots\!49}{20\!\cdots\!17}a-\frac{13\!\cdots\!53}{67\!\cdots\!17}$, $\frac{111044774058051}{14\!\cdots\!19}a^{10}+\frac{75875553644921}{14\!\cdots\!19}a^{9}-\frac{47\!\cdots\!32}{14\!\cdots\!19}a^{8}-\frac{29\!\cdots\!66}{14\!\cdots\!19}a^{7}+\frac{25\!\cdots\!81}{14\!\cdots\!19}a^{6}+\frac{14\!\cdots\!43}{14\!\cdots\!19}a^{5}-\frac{60\!\cdots\!39}{14\!\cdots\!19}a^{4}-\frac{39\!\cdots\!53}{29\!\cdots\!31}a^{3}+\frac{73\!\cdots\!06}{14\!\cdots\!19}a^{2}+\frac{11\!\cdots\!11}{29\!\cdots\!31}a-\frac{24\!\cdots\!61}{29\!\cdots\!31}$, $\frac{2545558285805}{29\!\cdots\!31}a^{10}-\frac{42138445585961}{14\!\cdots\!19}a^{9}-\frac{51\!\cdots\!33}{14\!\cdots\!19}a^{8}-\frac{42\!\cdots\!73}{20\!\cdots\!17}a^{7}+\frac{37\!\cdots\!04}{20\!\cdots\!17}a^{6}+\frac{26\!\cdots\!45}{20\!\cdots\!17}a^{5}-\frac{49\!\cdots\!30}{20\!\cdots\!17}a^{4}-\frac{78\!\cdots\!46}{33\!\cdots\!33}a^{3}-\frac{88\!\cdots\!02}{14\!\cdots\!19}a^{2}+\frac{21\!\cdots\!52}{20\!\cdots\!17}a+\frac{30\!\cdots\!78}{29\!\cdots\!31}$, $\frac{182484484833037}{14\!\cdots\!19}a^{10}+\frac{47292840548544}{20\!\cdots\!17}a^{9}-\frac{17\!\cdots\!11}{33\!\cdots\!33}a^{8}-\frac{58\!\cdots\!88}{14\!\cdots\!19}a^{7}+\frac{29\!\cdots\!47}{14\!\cdots\!19}a^{6}+\frac{30\!\cdots\!48}{14\!\cdots\!19}a^{5}-\frac{22\!\cdots\!16}{14\!\cdots\!19}a^{4}-\frac{49\!\cdots\!00}{14\!\cdots\!19}a^{3}-\frac{26\!\cdots\!69}{14\!\cdots\!19}a^{2}+\frac{33\!\cdots\!12}{20\!\cdots\!17}a+\frac{11\!\cdots\!59}{67\!\cdots\!17}$ Copy content Toggle raw display (assuming GRH)
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:  \( 1104123814020 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 1104123814020 \cdot 1}{2\cdot\sqrt{580095892065127629623589183049}}\cr\approx \mathstrut & 1.48445774033485 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 430*x^9 - 2035*x^8 + 28780*x^7 + 141142*x^6 - 760875*x^5 - 3028650*x^4 + 8674010*x^3 + 20981317*x^2 - 30077229*x - 47440673)
 
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 - 430*x^9 - 2035*x^8 + 28780*x^7 + 141142*x^6 - 760875*x^5 - 3028650*x^4 + 8674010*x^3 + 20981317*x^2 - 30077229*x - 47440673, 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 - 430*x^9 - 2035*x^8 + 28780*x^7 + 141142*x^6 - 760875*x^5 - 3028650*x^4 + 8674010*x^3 + 20981317*x^2 - 30077229*x - 47440673);
 
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 - 430*x^9 - 2035*x^8 + 28780*x^7 + 141142*x^6 - 760875*x^5 - 3028650*x^4 + 8674010*x^3 + 20981317*x^2 - 30077229*x - 47440673);
 
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

$C_{11}$ (as 11T1):

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 ${\href{/padicField/2.11.0.1}{11} }$ ${\href{/padicField/3.11.0.1}{11} }$ ${\href{/padicField/5.11.0.1}{11} }$ ${\href{/padicField/7.1.0.1}{1} }^{11}$ ${\href{/padicField/11.11.0.1}{11} }$ ${\href{/padicField/13.11.0.1}{11} }$ ${\href{/padicField/17.11.0.1}{11} }$ ${\href{/padicField/19.11.0.1}{11} }$ ${\href{/padicField/23.11.0.1}{11} }$ ${\href{/padicField/29.11.0.1}{11} }$ ${\href{/padicField/31.11.0.1}{11} }$ ${\href{/padicField/37.11.0.1}{11} }$ ${\href{/padicField/41.1.0.1}{1} }^{11}$ ${\href{/padicField/43.1.0.1}{1} }^{11}$ ${\href{/padicField/47.1.0.1}{1} }^{11}$ ${\href{/padicField/53.11.0.1}{11} }$ ${\href{/padicField/59.11.0.1}{11} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(947\) Copy content Toggle raw display Deg $11$$11$$1$$10$