Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 390*x^9 + 653*x^8 + 52046*x^7 - 146438*x^6 - 2723930*x^5 + 11558015*x^4 + 36326009*x^3 - 250960565*x^2 + 385923388*x - 145865807)
gp: K = bnfinit(y^11 - y^10 - 390*y^9 + 653*y^8 + 52046*y^7 - 146438*y^6 - 2723930*y^5 + 11558015*y^4 + 36326009*y^3 - 250960565*y^2 + 385923388*y - 145865807, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - x^10 - 390*x^9 + 653*x^8 + 52046*x^7 - 146438*x^6 - 2723930*x^5 + 11558015*x^4 + 36326009*x^3 - 250960565*x^2 + 385923388*x - 145865807);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^11 - x^10 - 390*x^9 + 653*x^8 + 52046*x^7 - 146438*x^6 - 2723930*x^5 + 11558015*x^4 + 36326009*x^3 - 250960565*x^2 + 385923388*x - 145865807)
\( x^{11} - x^{10} - 390 x^{9} + 653 x^{8} + 52046 x^{7} - 146438 x^{6} - 2723930 x^{5} + 11558015 x^{4} + \cdots - 145865807 \)
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: |
\(218741727855135890482344953401\)
\(\medspace = 859^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(464.80\) | 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: | $859^{10/11}\approx 464.8004208101595$ | ||
| Ramified primes: |
\(859\)
| 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: | \(859\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{859}(1,·)$, $\chi_{859}(61,·)$, $\chi_{859}(169,·)$, $\chi_{859}(13,·)$, $\chi_{859}(205,·)$, $\chi_{859}(269,·)$, $\chi_{859}(214,·)$, $\chi_{859}(88,·)$, $\chi_{859}(793,·)$, $\chi_{859}(285,·)$, $\chi_{859}(479,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11\!\cdots\!07}a^{10}+\frac{37\!\cdots\!69}{11\!\cdots\!07}a^{9}-\frac{10\!\cdots\!08}{11\!\cdots\!07}a^{8}-\frac{21\!\cdots\!52}{11\!\cdots\!07}a^{7}+\frac{71\!\cdots\!30}{11\!\cdots\!07}a^{6}-\frac{29\!\cdots\!20}{48\!\cdots\!09}a^{5}+\frac{39\!\cdots\!36}{64\!\cdots\!59}a^{4}+\frac{27\!\cdots\!34}{11\!\cdots\!07}a^{3}+\frac{12\!\cdots\!66}{11\!\cdots\!07}a^{2}+\frac{53\!\cdots\!12}{11\!\cdots\!07}a-\frac{36\!\cdots\!07}{11\!\cdots\!07}$
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: |
$\frac{57\!\cdots\!90}{11\!\cdots\!07}a^{10}+\frac{13\!\cdots\!34}{11\!\cdots\!07}a^{9}-\frac{19\!\cdots\!37}{11\!\cdots\!07}a^{8}-\frac{39\!\cdots\!94}{11\!\cdots\!07}a^{7}+\frac{20\!\cdots\!76}{11\!\cdots\!07}a^{6}+\frac{94\!\cdots\!93}{48\!\cdots\!09}a^{5}-\frac{44\!\cdots\!23}{64\!\cdots\!59}a^{4}+\frac{31\!\cdots\!09}{11\!\cdots\!07}a^{3}+\frac{93\!\cdots\!15}{11\!\cdots\!07}a^{2}-\frac{22\!\cdots\!58}{11\!\cdots\!07}a+\frac{17\!\cdots\!11}{11\!\cdots\!07}$, $\frac{29\!\cdots\!01}{11\!\cdots\!07}a^{10}-\frac{37\!\cdots\!31}{11\!\cdots\!07}a^{9}-\frac{11\!\cdots\!06}{11\!\cdots\!07}a^{8}+\frac{82\!\cdots\!47}{11\!\cdots\!07}a^{7}+\frac{15\!\cdots\!03}{11\!\cdots\!07}a^{6}-\frac{11\!\cdots\!32}{48\!\cdots\!09}a^{5}-\frac{50\!\cdots\!96}{64\!\cdots\!59}a^{4}+\frac{24\!\cdots\!28}{11\!\cdots\!07}a^{3}+\frac{14\!\cdots\!79}{11\!\cdots\!07}a^{2}-\frac{56\!\cdots\!74}{11\!\cdots\!07}a+\frac{47\!\cdots\!63}{11\!\cdots\!07}$, $\frac{29\!\cdots\!45}{11\!\cdots\!07}a^{10}+\frac{22\!\cdots\!86}{11\!\cdots\!07}a^{9}-\frac{11\!\cdots\!60}{11\!\cdots\!07}a^{8}-\frac{22\!\cdots\!00}{11\!\cdots\!07}a^{7}+\frac{14\!\cdots\!04}{11\!\cdots\!07}a^{6}-\frac{53\!\cdots\!55}{48\!\cdots\!09}a^{5}-\frac{45\!\cdots\!22}{64\!\cdots\!59}a^{4}+\frac{15\!\cdots\!40}{11\!\cdots\!07}a^{3}+\frac{13\!\cdots\!09}{11\!\cdots\!07}a^{2}-\frac{38\!\cdots\!74}{11\!\cdots\!07}a+\frac{16\!\cdots\!85}{11\!\cdots\!07}$, $\frac{65\!\cdots\!83}{48\!\cdots\!09}a^{10}+\frac{10\!\cdots\!43}{48\!\cdots\!09}a^{9}-\frac{25\!\cdots\!42}{48\!\cdots\!09}a^{8}-\frac{22\!\cdots\!26}{48\!\cdots\!09}a^{7}+\frac{33\!\cdots\!47}{48\!\cdots\!09}a^{6}-\frac{91\!\cdots\!97}{48\!\cdots\!09}a^{5}-\frac{10\!\cdots\!91}{28\!\cdots\!33}a^{4}+\frac{28\!\cdots\!21}{48\!\cdots\!09}a^{3}+\frac{31\!\cdots\!02}{48\!\cdots\!09}a^{2}-\frac{82\!\cdots\!29}{48\!\cdots\!09}a+\frac{36\!\cdots\!60}{48\!\cdots\!09}$, $\frac{72\!\cdots\!60}{11\!\cdots\!07}a^{10}+\frac{75\!\cdots\!77}{11\!\cdots\!07}a^{9}-\frac{26\!\cdots\!31}{11\!\cdots\!07}a^{8}-\frac{17\!\cdots\!48}{11\!\cdots\!07}a^{7}+\frac{34\!\cdots\!23}{11\!\cdots\!07}a^{6}-\frac{50\!\cdots\!01}{48\!\cdots\!09}a^{5}-\frac{10\!\cdots\!54}{64\!\cdots\!59}a^{4}+\frac{28\!\cdots\!51}{11\!\cdots\!07}a^{3}+\frac{30\!\cdots\!87}{11\!\cdots\!07}a^{2}-\frac{83\!\cdots\!55}{11\!\cdots\!07}a+\frac{42\!\cdots\!89}{11\!\cdots\!07}$, $\frac{22\!\cdots\!24}{48\!\cdots\!09}a^{10}+\frac{15\!\cdots\!33}{48\!\cdots\!09}a^{9}-\frac{86\!\cdots\!55}{48\!\cdots\!09}a^{8}-\frac{19\!\cdots\!50}{48\!\cdots\!09}a^{7}+\frac{11\!\cdots\!46}{48\!\cdots\!09}a^{6}-\frac{80\!\cdots\!64}{48\!\cdots\!09}a^{5}-\frac{34\!\cdots\!29}{28\!\cdots\!33}a^{4}+\frac{10\!\cdots\!03}{48\!\cdots\!09}a^{3}+\frac{10\!\cdots\!24}{48\!\cdots\!09}a^{2}-\frac{28\!\cdots\!71}{48\!\cdots\!09}a+\frac{10\!\cdots\!57}{48\!\cdots\!09}$, $\frac{74\!\cdots\!67}{11\!\cdots\!07}a^{10}+\frac{31\!\cdots\!06}{11\!\cdots\!07}a^{9}-\frac{31\!\cdots\!63}{11\!\cdots\!07}a^{8}-\frac{74\!\cdots\!75}{11\!\cdots\!07}a^{7}+\frac{43\!\cdots\!08}{11\!\cdots\!07}a^{6}+\frac{70\!\cdots\!20}{48\!\cdots\!09}a^{5}-\frac{14\!\cdots\!20}{64\!\cdots\!59}a^{4}+\frac{37\!\cdots\!48}{11\!\cdots\!07}a^{3}+\frac{43\!\cdots\!36}{11\!\cdots\!07}a^{2}-\frac{12\!\cdots\!53}{11\!\cdots\!07}a+\frac{55\!\cdots\!48}{11\!\cdots\!07}$, $\frac{10\!\cdots\!22}{11\!\cdots\!07}a^{10}+\frac{69\!\cdots\!11}{11\!\cdots\!07}a^{9}-\frac{35\!\cdots\!47}{11\!\cdots\!07}a^{8}-\frac{21\!\cdots\!38}{11\!\cdots\!07}a^{7}+\frac{37\!\cdots\!88}{11\!\cdots\!07}a^{6}+\frac{75\!\cdots\!01}{48\!\cdots\!09}a^{5}-\frac{88\!\cdots\!28}{64\!\cdots\!59}a^{4}-\frac{30\!\cdots\!69}{11\!\cdots\!07}a^{3}+\frac{21\!\cdots\!31}{11\!\cdots\!07}a^{2}-\frac{23\!\cdots\!02}{11\!\cdots\!07}a+\frac{78\!\cdots\!06}{11\!\cdots\!07}$, $\frac{24\!\cdots\!22}{11\!\cdots\!07}a^{10}+\frac{18\!\cdots\!89}{11\!\cdots\!07}a^{9}-\frac{98\!\cdots\!06}{11\!\cdots\!07}a^{8}-\frac{52\!\cdots\!28}{11\!\cdots\!07}a^{7}+\frac{14\!\cdots\!90}{11\!\cdots\!07}a^{6}+\frac{14\!\cdots\!50}{48\!\cdots\!09}a^{5}-\frac{47\!\cdots\!13}{64\!\cdots\!59}a^{4}+\frac{49\!\cdots\!17}{11\!\cdots\!07}a^{3}+\frac{14\!\cdots\!01}{11\!\cdots\!07}a^{2}-\frac{42\!\cdots\!28}{11\!\cdots\!07}a+\frac{35\!\cdots\!43}{11\!\cdots\!07}$, $\frac{12\!\cdots\!27}{11\!\cdots\!07}a^{10}-\frac{12\!\cdots\!07}{11\!\cdots\!07}a^{9}-\frac{44\!\cdots\!39}{11\!\cdots\!07}a^{8}+\frac{45\!\cdots\!40}{11\!\cdots\!07}a^{7}+\frac{45\!\cdots\!11}{11\!\cdots\!07}a^{6}-\frac{24\!\cdots\!02}{48\!\cdots\!09}a^{5}-\frac{30\!\cdots\!29}{64\!\cdots\!59}a^{4}+\frac{22\!\cdots\!81}{11\!\cdots\!07}a^{3}-\frac{82\!\cdots\!02}{11\!\cdots\!07}a^{2}+\frac{10\!\cdots\!42}{11\!\cdots\!07}a-\frac{38\!\cdots\!57}{11\!\cdots\!07}$
| 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: | \( 80664010957.6 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 80664010957.6 \cdot 1}{2\cdot\sqrt{218741727855135890482344953401}}\cr\approx \mathstrut & 0.176609455129 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 390*x^9 + 653*x^8 + 52046*x^7 - 146438*x^6 - 2723930*x^5 + 11558015*x^4 + 36326009*x^3 - 250960565*x^2 + 385923388*x - 145865807)
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 - 390*x^9 + 653*x^8 + 52046*x^7 - 146438*x^6 - 2723930*x^5 + 11558015*x^4 + 36326009*x^3 - 250960565*x^2 + 385923388*x - 145865807, 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 - 390*x^9 + 653*x^8 + 52046*x^7 - 146438*x^6 - 2723930*x^5 + 11558015*x^4 + 36326009*x^3 - 250960565*x^2 + 385923388*x - 145865807);
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 - 390*x^9 + 653*x^8 + 52046*x^7 - 146438*x^6 - 2723930*x^5 + 11558015*x^4 + 36326009*x^3 - 250960565*x^2 + 385923388*x - 145865807);
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 | ${\href{/padicField/2.11.0.1}{11} }$ | ${\href{/padicField/3.11.0.1}{11} }$ | ${\href{/padicField/5.11.0.1}{11} }$ | ${\href{/padicField/7.11.0.1}{11} }$ | ${\href{/padicField/11.11.0.1}{11} }$ | ${\href{/padicField/13.11.0.1}{11} }$ | ${\href{/padicField/17.11.0.1}{11} }$ | ${\href{/padicField/19.11.0.1}{11} }$ | ${\href{/padicField/23.1.0.1}{1} }^{11}$ | ${\href{/padicField/29.11.0.1}{11} }$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.11.0.1}{11} }$ | ${\href{/padicField/41.11.0.1}{11} }$ | ${\href{/padicField/43.11.0.1}{11} }$ | ${\href{/padicField/47.11.0.1}{11} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.11.0.1}{11} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(859\)
| Deg $11$ | $11$ | $1$ | $10$ |