Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 13*x^20 + 76*x^18 - 269*x^16 + 659*x^14 - 1206*x^12 + 1700*x^10 - 1829*x^8 + 1455*x^6 - 817*x^4 + 293*x^2 - 49)
gp: K = bnfinit(y^22 - 13*y^20 + 76*y^18 - 269*y^16 + 659*y^14 - 1206*y^12 + 1700*y^10 - 1829*y^8 + 1455*y^6 - 817*y^4 + 293*y^2 - 49, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 13*x^20 + 76*x^18 - 269*x^16 + 659*x^14 - 1206*x^12 + 1700*x^10 - 1829*x^8 + 1455*x^6 - 817*x^4 + 293*x^2 - 49);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 13*x^20 + 76*x^18 - 269*x^16 + 659*x^14 - 1206*x^12 + 1700*x^10 - 1829*x^8 + 1455*x^6 - 817*x^4 + 293*x^2 - 49)
\( x^{22} - 13 x^{20} + 76 x^{18} - 269 x^{16} + 659 x^{14} - 1206 x^{12} + 1700 x^{10} - 1829 x^{8} + \cdots - 49 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[6, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(529866739430089519756710630129664\)
\(\medspace = 2^{22}\cdot 1831^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.72\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(1831\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{839}a^{20}+\frac{395}{839}a^{18}+\frac{148}{839}a^{16}-\frac{293}{839}a^{14}+\frac{253}{839}a^{12}-\frac{340}{839}a^{10}-\frac{263}{839}a^{8}-\frac{63}{839}a^{6}+\frac{82}{839}a^{4}-\frac{82}{839}a^{2}+\frac{397}{839}$, $\frac{1}{5873}a^{21}+\frac{2073}{5873}a^{19}+\frac{1826}{5873}a^{17}-\frac{2810}{5873}a^{15}+\frac{253}{5873}a^{13}-\frac{2018}{5873}a^{11}-\frac{2780}{5873}a^{9}+\frac{1615}{5873}a^{7}-\frac{757}{5873}a^{5}-\frac{82}{5873}a^{3}-\frac{442}{5873}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| 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: | $13$ | 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: |
$a^{2}-1$, $\frac{576}{839}a^{20}-\frac{6561}{839}a^{18}+\frac{33230}{839}a^{16}-\frac{101648}{839}a^{14}+\frac{217043}{839}a^{12}-\frac{348538}{839}a^{10}+\frac{424066}{839}a^{8}-\frac{377761}{839}a^{6}+\frac{234329}{839}a^{4}-\frac{91699}{839}a^{2}+\frac{15566}{839}$, $\frac{5435}{5873}a^{21}-\frac{62262}{5873}a^{19}+\frac{316082}{5873}a^{17}-\frac{965722}{5873}a^{15}+\frac{2056323}{5873}a^{13}-\frac{3297692}{5873}a^{11}+\frac{4013188}{5873}a^{9}-\frac{3579267}{5873}a^{7}+\frac{2246164}{5873}a^{5}-\frac{915510}{5873}a^{3}+\frac{170104}{5873}a$, $\frac{429}{5873}a^{21}-\frac{3379}{5873}a^{19}+\frac{8118}{5873}a^{17}+\frac{4348}{5873}a^{15}-\frac{67653}{5873}a^{13}+\frac{197291}{5873}a^{11}-\frac{370400}{5873}a^{9}+\frac{487280}{5873}a^{7}-\frac{430467}{5873}a^{5}+\frac{252599}{5873}a^{3}-\frac{89777}{5873}a$, $\frac{1146}{5873}a^{21}-\frac{14653}{5873}a^{19}+\frac{84030}{5873}a^{17}-\frac{289633}{5873}a^{15}+\frac{683429}{5873}a^{13}-\frac{1190885}{5873}a^{11}+\frac{1577123}{5873}a^{9}-\frac{1549677}{5873}a^{7}+\frac{1058822}{5873}a^{5}-\frac{458098}{5873}a^{3}+\frac{98387}{5873}a$, $\frac{2958}{5873}a^{21}-\frac{34716}{5873}a^{19}+\frac{180211}{5873}a^{17}-\frac{559620}{5873}a^{15}+\frac{1200595}{5873}a^{13}-\frac{1928620}{5873}a^{11}+\frac{2348160}{5873}a^{9}-\frac{2082494}{5873}a^{7}+\frac{1278721}{5873}a^{5}-\frac{500968}{5873}a^{3}+\frac{84465}{5873}a$, $a+1$, $\frac{429}{5873}a^{21}-\frac{404}{839}a^{20}-\frac{3379}{5873}a^{19}+\frac{4864}{839}a^{18}+\frac{8118}{5873}a^{17}-\frac{26232}{839}a^{16}+\frac{4348}{5873}a^{15}+\frac{85651}{839}a^{14}-\frac{67653}{5873}a^{13}-\frac{194502}{839}a^{12}+\frac{197291}{5873}a^{11}+\frac{331169}{839}a^{10}-\frac{370400}{5873}a^{9}-\frac{430708}{839}a^{8}+\frac{487280}{5873}a^{7}+\frac{418104}{839}a^{6}-\frac{430467}{5873}a^{5}-\frac{289862}{839}a^{4}+\frac{252599}{5873}a^{3}+\frac{132130}{839}a^{2}-\frac{89777}{5873}a-\frac{30343}{839}$, $\frac{838}{5873}a^{21}-\frac{674}{839}a^{20}-\frac{7107}{5873}a^{19}+\frac{8123}{839}a^{18}+\frac{20827}{5873}a^{17}-\frac{43539}{839}a^{16}-\frac{11453}{5873}a^{15}+\frac{140430}{839}a^{14}-\frac{87509}{5873}a^{13}-\frac{313991}{839}a^{12}+\frac{317482}{5873}a^{11}+\frac{526166}{839}a^{10}-\frac{644089}{5873}a^{9}-\frac{672645}{839}a^{8}+\frac{889403}{5873}a^{7}+\frac{638152}{839}a^{6}-\frac{816429}{5873}a^{5}-\frac{429462}{839}a^{4}+\frac{471600}{5873}a^{3}+\frac{190347}{839}a^{2}-\frac{147222}{5873}a-\frac{42726}{839}$, $\frac{1849}{5873}a^{21}-\frac{558}{839}a^{20}-\frac{19711}{5873}a^{19}+\frac{6120}{839}a^{18}+\frac{93267}{5873}a^{17}-\frac{29727}{839}a^{16}-\frac{268243}{5873}a^{15}+\frac{87145}{839}a^{14}+\frac{544146}{5873}a^{13}-\frac{178929}{839}a^{12}-\frac{830020}{5873}a^{11}+\frac{276976}{839}a^{10}+\frac{944208}{5873}a^{9}-\frac{323086}{839}a^{8}-\frac{772585}{5873}a^{7}+\frac{273430}{839}a^{6}+\frac{420937}{5873}a^{5}-\frac{160699}{839}a^{4}-\frac{122253}{5873}a^{3}+\frac{60858}{839}a^{2}+\frac{4962}{5873}a-\frac{10937}{839}$, $\frac{654}{5873}a^{21}+\frac{576}{839}a^{20}-\frac{6794}{5873}a^{19}-\frac{6561}{839}a^{18}+\frac{31350}{5873}a^{17}+\frac{33230}{839}a^{16}-\frac{87586}{5873}a^{15}-\frac{101648}{839}a^{14}+\frac{171335}{5873}a^{13}+\frac{217043}{839}a^{12}-\frac{250886}{5873}a^{11}-\frac{348538}{839}a^{10}+\frac{272668}{5873}a^{9}+\frac{424066}{839}a^{8}-\frac{206485}{5873}a^{7}-\frac{377761}{839}a^{6}+\frac{92222}{5873}a^{5}+\frac{235168}{839}a^{4}-\frac{12517}{5873}a^{3}-\frac{94216}{839}a^{2}-\frac{7164}{5873}a+\frac{17244}{839}$, $\frac{11308}{5873}a^{21}+\frac{1604}{839}a^{20}-\frac{132738}{5873}a^{19}-\frac{18323}{839}a^{18}+\frac{691954}{5873}a^{17}+\frac{93084}{839}a^{16}-\frac{2169687}{5873}a^{15}-\frac{285392}{839}a^{14}+\frac{4722665}{5873}a^{13}+\frac{609689}{839}a^{12}-\frac{7714188}{5873}a^{11}-\frac{978284}{839}a^{10}+\frac{9580792}{5873}a^{9}+\frac{1189867}{839}a^{8}-\frac{8753380}{5873}a^{7}-\frac{1060029}{839}a^{6}+\frac{5617266}{5873}a^{5}+\frac{657581}{839}a^{4}-\frac{2336776}{5873}a^{3}-\frac{260734}{839}a^{2}+\frac{457881}{5873}a+\frac{47810}{839}$, $\frac{1697}{5873}a^{21}+\frac{376}{839}a^{20}-\frac{17665}{5873}a^{19}-\frac{4178}{839}a^{18}+\frac{80000}{5873}a^{17}+\frac{20410}{839}a^{16}-\frac{216995}{5873}a^{15}-\frac{59828}{839}a^{14}+\frac{417595}{5873}a^{13}+\frac{122815}{839}a^{12}-\frac{617252}{5873}a^{11}-\frac{190765}{839}a^{10}+\frac{685500}{5873}a^{9}+\frac{223288}{839}a^{8}-\frac{554098}{5873}a^{7}-\frac{189810}{839}a^{6}+\frac{330446}{5873}a^{5}+\frac{114732}{839}a^{4}-\frac{133281}{5873}a^{3}-\frac{45934}{839}a^{2}+\frac{25162}{5873}a+\frac{9159}{839}$
| 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: | \( 81981104.2428 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot 81981104.2428 \cdot 1}{2\cdot\sqrt{529866739430089519756710630129664}}\cr\approx \mathstrut & 0.276833956327 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 13*x^20 + 76*x^18 - 269*x^16 + 659*x^14 - 1206*x^12 + 1700*x^10 - 1829*x^8 + 1455*x^6 - 817*x^4 + 293*x^2 - 49)
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^22 - 13*x^20 + 76*x^18 - 269*x^16 + 659*x^14 - 1206*x^12 + 1700*x^10 - 1829*x^8 + 1455*x^6 - 817*x^4 + 293*x^2 - 49, 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^22 - 13*x^20 + 76*x^18 - 269*x^16 + 659*x^14 - 1206*x^12 + 1700*x^10 - 1829*x^8 + 1455*x^6 - 817*x^4 + 293*x^2 - 49);
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^22 - 13*x^20 + 76*x^18 - 269*x^16 + 659*x^14 - 1206*x^12 + 1700*x^10 - 1829*x^8 + 1455*x^6 - 817*x^4 + 293*x^2 - 49);
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
$C_2^{10}.\PSL(2,11)$ (as 22T39):
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 675840 |
| The 56 conjugacy class representatives for $C_2^{10}.\PSL(2,11)$ |
| Character table for $C_2^{10}.\PSL(2,11)$ |
Intermediate fields
| 11.3.11239665258721.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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]
Sibling fields
| Degree 22 sibling: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.11.0.1}{11} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }^{5}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.11.0.1}{11} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $22$ | $2$ | $11$ | $22$ | |||
|
\(1831\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $12$ | $2$ | $6$ | $6$ |