Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 1265*x^9 - 4807*x^8 + 544962*x^7 + 4031808*x^6 - 81455880*x^5 - 916234187*x^4 + 714773070*x^3 + 31567681332*x^2 + 58422539890*x - 98108643071)
gp: K = bnfinit(y^11 - 1265*y^9 - 4807*y^8 + 544962*y^7 + 4031808*y^6 - 81455880*y^5 - 916234187*y^4 + 714773070*y^3 + 31567681332*y^2 + 58422539890*y - 98108643071, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 1265*x^9 - 4807*x^8 + 544962*x^7 + 4031808*x^6 - 81455880*x^5 - 916234187*x^4 + 714773070*x^3 + 31567681332*x^2 + 58422539890*x - 98108643071);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^11 - 1265*x^9 - 4807*x^8 + 544962*x^7 + 4031808*x^6 - 81455880*x^5 - 916234187*x^4 + 714773070*x^3 + 31567681332*x^2 + 58422539890*x - 98108643071)
\( x^{11} - 1265 x^{9} - 4807 x^{8} + 544962 x^{7} + 4031808 x^{6} - 81455880 x^{5} - 916234187 x^{4} + \cdots - 98108643071 \)
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: |
\(27869685209056005146671841116784449\)
\(\medspace = 11^{20}\cdot 23^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(1353.24\) | 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: | $11^{20/11}23^{10/11}\approx 1353.24254149841$ | ||
| Ramified primes: |
\(11\), \(23\)
| 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: | \(2783=11^{2}\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2783}(1,·)$, $\chi_{2783}(1442,·)$, $\chi_{2783}(1156,·)$, $\chi_{2783}(518,·)$, $\chi_{2783}(2509,·)$, $\chi_{2783}(78,·)$, $\chi_{2783}(463,·)$, $\chi_{2783}(496,·)$, $\chi_{2783}(1112,·)$, $\chi_{2783}(892,·)$, $\chi_{2783}(2718,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{27\!\cdots\!79}a^{10}-\frac{76\!\cdots\!03}{27\!\cdots\!79}a^{9}+\frac{10\!\cdots\!98}{27\!\cdots\!79}a^{8}-\frac{62\!\cdots\!33}{27\!\cdots\!79}a^{7}+\frac{60\!\cdots\!22}{27\!\cdots\!79}a^{6}-\frac{51\!\cdots\!18}{27\!\cdots\!79}a^{5}-\frac{64\!\cdots\!45}{27\!\cdots\!79}a^{4}-\frac{68\!\cdots\!15}{27\!\cdots\!79}a^{3}+\frac{11\!\cdots\!82}{27\!\cdots\!79}a^{2}+\frac{15\!\cdots\!10}{27\!\cdots\!79}a-\frac{50\!\cdots\!27}{72\!\cdots\!13}$
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
$C_{11}$, which has order $11$ (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: |
$\frac{18\!\cdots\!20}{27\!\cdots\!79}a^{10}-\frac{14\!\cdots\!54}{27\!\cdots\!79}a^{9}-\frac{22\!\cdots\!66}{27\!\cdots\!79}a^{8}+\frac{87\!\cdots\!44}{27\!\cdots\!79}a^{7}+\frac{92\!\cdots\!28}{27\!\cdots\!79}a^{6}-\frac{14\!\cdots\!71}{27\!\cdots\!79}a^{5}-\frac{14\!\cdots\!40}{27\!\cdots\!79}a^{4}-\frac{49\!\cdots\!19}{27\!\cdots\!79}a^{3}+\frac{52\!\cdots\!10}{27\!\cdots\!79}a^{2}+\frac{16\!\cdots\!99}{27\!\cdots\!79}a-\frac{58\!\cdots\!09}{72\!\cdots\!13}$, $\frac{10\!\cdots\!57}{27\!\cdots\!79}a^{10}+\frac{52\!\cdots\!35}{27\!\cdots\!79}a^{9}-\frac{10\!\cdots\!87}{27\!\cdots\!79}a^{8}-\frac{70\!\cdots\!13}{27\!\cdots\!79}a^{7}+\frac{30\!\cdots\!31}{27\!\cdots\!79}a^{6}+\frac{23\!\cdots\!97}{27\!\cdots\!79}a^{5}-\frac{26\!\cdots\!24}{27\!\cdots\!79}a^{4}-\frac{16\!\cdots\!33}{27\!\cdots\!79}a^{3}+\frac{75\!\cdots\!55}{27\!\cdots\!79}a^{2}+\frac{30\!\cdots\!89}{27\!\cdots\!79}a-\frac{10\!\cdots\!66}{72\!\cdots\!13}$, $\frac{48\!\cdots\!22}{27\!\cdots\!79}a^{10}-\frac{21\!\cdots\!13}{27\!\cdots\!79}a^{9}-\frac{60\!\cdots\!85}{27\!\cdots\!79}a^{8}+\frac{32\!\cdots\!82}{27\!\cdots\!79}a^{7}+\frac{26\!\cdots\!41}{27\!\cdots\!79}a^{6}+\frac{79\!\cdots\!30}{27\!\cdots\!79}a^{5}-\frac{42\!\cdots\!01}{27\!\cdots\!79}a^{4}-\frac{25\!\cdots\!86}{27\!\cdots\!79}a^{3}+\frac{14\!\cdots\!46}{27\!\cdots\!79}a^{2}+\frac{88\!\cdots\!03}{27\!\cdots\!79}a-\frac{28\!\cdots\!89}{72\!\cdots\!13}$, $\frac{61\!\cdots\!95}{27\!\cdots\!79}a^{10}+\frac{88\!\cdots\!28}{27\!\cdots\!79}a^{9}-\frac{57\!\cdots\!80}{27\!\cdots\!79}a^{8}-\frac{10\!\cdots\!10}{27\!\cdots\!79}a^{7}+\frac{10\!\cdots\!39}{27\!\cdots\!79}a^{6}+\frac{31\!\cdots\!60}{27\!\cdots\!79}a^{5}+\frac{15\!\cdots\!44}{27\!\cdots\!79}a^{4}-\frac{71\!\cdots\!74}{27\!\cdots\!79}a^{3}-\frac{70\!\cdots\!21}{27\!\cdots\!79}a^{2}-\frac{91\!\cdots\!18}{27\!\cdots\!79}a+\frac{48\!\cdots\!73}{72\!\cdots\!13}$, $\frac{12\!\cdots\!91}{27\!\cdots\!79}a^{10}-\frac{28\!\cdots\!86}{27\!\cdots\!79}a^{9}-\frac{94\!\cdots\!13}{27\!\cdots\!79}a^{8}+\frac{18\!\cdots\!72}{27\!\cdots\!79}a^{7}+\frac{28\!\cdots\!39}{27\!\cdots\!79}a^{6}-\frac{32\!\cdots\!14}{27\!\cdots\!79}a^{5}-\frac{41\!\cdots\!67}{27\!\cdots\!79}a^{4}+\frac{94\!\cdots\!61}{27\!\cdots\!79}a^{3}+\frac{12\!\cdots\!09}{27\!\cdots\!79}a^{2}-\frac{33\!\cdots\!42}{27\!\cdots\!79}a-\frac{30\!\cdots\!94}{72\!\cdots\!13}$, $\frac{17\!\cdots\!67}{27\!\cdots\!79}a^{10}-\frac{16\!\cdots\!98}{27\!\cdots\!79}a^{9}-\frac{20\!\cdots\!12}{27\!\cdots\!79}a^{8}+\frac{13\!\cdots\!56}{27\!\cdots\!79}a^{7}+\frac{83\!\cdots\!41}{27\!\cdots\!79}a^{6}-\frac{19\!\cdots\!49}{27\!\cdots\!79}a^{5}-\frac{12\!\cdots\!55}{27\!\cdots\!79}a^{4}-\frac{24\!\cdots\!19}{27\!\cdots\!79}a^{3}+\frac{40\!\cdots\!53}{27\!\cdots\!79}a^{2}+\frac{10\!\cdots\!71}{27\!\cdots\!79}a-\frac{40\!\cdots\!98}{72\!\cdots\!13}$, $\frac{21\!\cdots\!80}{27\!\cdots\!79}a^{10}+\frac{43\!\cdots\!50}{27\!\cdots\!79}a^{9}-\frac{19\!\cdots\!62}{27\!\cdots\!79}a^{8}-\frac{49\!\cdots\!19}{27\!\cdots\!79}a^{7}+\frac{22\!\cdots\!18}{27\!\cdots\!79}a^{6}+\frac{13\!\cdots\!27}{27\!\cdots\!79}a^{5}+\frac{10\!\cdots\!47}{27\!\cdots\!79}a^{4}-\frac{15\!\cdots\!51}{27\!\cdots\!79}a^{3}-\frac{33\!\cdots\!36}{27\!\cdots\!79}a^{2}-\frac{55\!\cdots\!70}{27\!\cdots\!79}a+\frac{26\!\cdots\!79}{72\!\cdots\!13}$, $\frac{35\!\cdots\!04}{27\!\cdots\!79}a^{10}-\frac{66\!\cdots\!05}{27\!\cdots\!79}a^{9}-\frac{32\!\cdots\!33}{27\!\cdots\!79}a^{8}+\frac{44\!\cdots\!53}{27\!\cdots\!79}a^{7}+\frac{11\!\cdots\!03}{27\!\cdots\!79}a^{6}-\frac{69\!\cdots\!94}{27\!\cdots\!79}a^{5}-\frac{16\!\cdots\!56}{27\!\cdots\!79}a^{4}-\frac{91\!\cdots\!83}{27\!\cdots\!79}a^{3}+\frac{55\!\cdots\!30}{27\!\cdots\!79}a^{2}+\frac{10\!\cdots\!11}{27\!\cdots\!79}a-\frac{66\!\cdots\!82}{72\!\cdots\!13}$, $\frac{17\!\cdots\!13}{27\!\cdots\!79}a^{10}-\frac{15\!\cdots\!82}{27\!\cdots\!79}a^{9}-\frac{20\!\cdots\!85}{27\!\cdots\!79}a^{8}+\frac{10\!\cdots\!86}{27\!\cdots\!79}a^{7}+\frac{83\!\cdots\!93}{27\!\cdots\!79}a^{6}-\frac{82\!\cdots\!29}{27\!\cdots\!79}a^{5}-\frac{13\!\cdots\!04}{27\!\cdots\!79}a^{4}-\frac{35\!\cdots\!55}{27\!\cdots\!79}a^{3}+\frac{44\!\cdots\!46}{27\!\cdots\!79}a^{2}+\frac{12\!\cdots\!44}{27\!\cdots\!79}a-\frac{47\!\cdots\!15}{72\!\cdots\!13}$, $\frac{11\!\cdots\!80}{27\!\cdots\!79}a^{10}+\frac{13\!\cdots\!44}{27\!\cdots\!79}a^{9}-\frac{14\!\cdots\!80}{27\!\cdots\!79}a^{8}-\frac{72\!\cdots\!88}{27\!\cdots\!79}a^{7}+\frac{62\!\cdots\!91}{27\!\cdots\!79}a^{6}+\frac{53\!\cdots\!28}{27\!\cdots\!79}a^{5}-\frac{88\!\cdots\!42}{27\!\cdots\!79}a^{4}-\frac{11\!\cdots\!21}{27\!\cdots\!79}a^{3}-\frac{45\!\cdots\!71}{27\!\cdots\!79}a^{2}+\frac{36\!\cdots\!62}{27\!\cdots\!79}a+\frac{28\!\cdots\!09}{72\!\cdots\!13}$
| 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: | \( 2524169127627.655 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 2524169127627.655 \cdot 11}{2\cdot\sqrt{27869685209056005146671841116784449}}\cr\approx \mathstrut & 0.170311926223595 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^11 - 1265*x^9 - 4807*x^8 + 544962*x^7 + 4031808*x^6 - 81455880*x^5 - 916234187*x^4 + 714773070*x^3 + 31567681332*x^2 + 58422539890*x - 98108643071)
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 - 1265*x^9 - 4807*x^8 + 544962*x^7 + 4031808*x^6 - 81455880*x^5 - 916234187*x^4 + 714773070*x^3 + 31567681332*x^2 + 58422539890*x - 98108643071, 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 - 1265*x^9 - 4807*x^8 + 544962*x^7 + 4031808*x^6 - 81455880*x^5 - 916234187*x^4 + 714773070*x^3 + 31567681332*x^2 + 58422539890*x - 98108643071);
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 - 1265*x^9 - 4807*x^8 + 544962*x^7 + 4031808*x^6 - 81455880*x^5 - 916234187*x^4 + 714773070*x^3 + 31567681332*x^2 + 58422539890*x - 98108643071);
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 | ${\href{/padicField/2.11.0.1}{11} }$ | ${\href{/padicField/3.11.0.1}{11} }$ | ${\href{/padicField/5.11.0.1}{11} }$ | ${\href{/padicField/7.11.0.1}{11} }$ | R | ${\href{/padicField/13.11.0.1}{11} }$ | ${\href{/padicField/17.11.0.1}{11} }$ | ${\href{/padicField/19.11.0.1}{11} }$ | R | ${\href{/padicField/29.11.0.1}{11} }$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.11.0.1}{11} }$ | ${\href{/padicField/41.11.0.1}{11} }$ | ${\href{/padicField/43.11.0.1}{11} }$ | ${\href{/padicField/47.11.0.1}{11} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.11.0.1}{11} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.11.20.7 | $x^{11} + 110 x^{10} + 1100$ | $11$ | $1$ | $20$ | $C_{11}$ | $[2]$ |
|
\(23\)
| 23.11.10.9 | $x^{11} + 46$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |