Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 44*x^16 - 11*x^14 - 88*x^12 + 264*x^10 - 275*x^8 + 154*x^6 - 55*x^4 + 11*x^2 - 1)
gp: K = bnfinit(y^22 - 44*y^16 - 11*y^14 - 88*y^12 + 264*y^10 - 275*y^8 + 154*y^6 - 55*y^4 + 11*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 44*x^16 - 11*x^14 - 88*x^12 + 264*x^10 - 275*x^8 + 154*x^6 - 55*x^4 + 11*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 44*x^16 - 11*x^14 - 88*x^12 + 264*x^10 - 275*x^8 + 154*x^6 - 55*x^4 + 11*x^2 - 1)
\( x^{22} - 44x^{16} - 11x^{14} - 88x^{12} + 264x^{10} - 275x^{8} + 154x^{6} - 55x^{4} + 11x^{2} - 1 \)
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: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1249711388122384634961076169998336\)
\(\medspace = 2^{20}\cdot 11^{26}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.94\) | 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: |
\(2\), \(11\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{53887063}a^{20}+\frac{19623235}{53887063}a^{18}+\frac{3921777}{53887063}a^{16}+\frac{4858109}{53887063}a^{14}-\frac{4138948}{53887063}a^{12}+\frac{2063866}{53887063}a^{10}-\frac{10750947}{53887063}a^{8}-\frac{22825253}{53887063}a^{6}-\frac{23539963}{53887063}a^{4}+\frac{24511736}{53887063}a^{2}+\frac{21014750}{53887063}$, $\frac{1}{107774126}a^{21}-\frac{1}{107774126}a^{20}+\frac{19623235}{107774126}a^{19}-\frac{19623235}{107774126}a^{18}+\frac{3921777}{107774126}a^{17}-\frac{3921777}{107774126}a^{16}+\frac{4858109}{107774126}a^{15}-\frac{4858109}{107774126}a^{14}-\frac{2069474}{53887063}a^{13}+\frac{2069474}{53887063}a^{12}+\frac{1031933}{53887063}a^{11}-\frac{1031933}{53887063}a^{10}+\frac{21568058}{53887063}a^{9}-\frac{21568058}{53887063}a^{8}-\frac{22825253}{107774126}a^{7}+\frac{22825253}{107774126}a^{6}-\frac{23539963}{107774126}a^{5}+\frac{23539963}{107774126}a^{4}+\frac{12255868}{53887063}a^{3}-\frac{12255868}{53887063}a^{2}-\frac{32872313}{107774126}a+\frac{32872313}{107774126}$
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: | $11$ | 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: |
$a$, $\frac{13818834}{53887063}a^{21}+\frac{18014705}{53887063}a^{19}+\frac{4427729}{53887063}a^{17}-\frac{610098258}{53887063}a^{15}-\frac{946080881}{53887063}a^{13}-\frac{1608035963}{53887063}a^{11}+\frac{2104609399}{53887063}a^{9}+\frac{649690103}{53887063}a^{7}-\frac{1489846854}{53887063}a^{5}+\frac{364117487}{53887063}a^{3}+\frac{19360232}{53887063}a$, $\frac{589550495}{107774126}a^{21}+\frac{286319127}{107774126}a^{20}+\frac{124686367}{107774126}a^{19}+\frac{49500597}{107774126}a^{18}+\frac{25685173}{107774126}a^{17}+\frac{8606729}{107774126}a^{16}-\frac{25931750065}{107774126}a^{15}-\frac{12595180739}{107774126}a^{14}-\frac{5984252788}{53887063}a^{13}-\frac{2663223523}{53887063}a^{12}-\frac{27190785215}{53887063}a^{11}-\frac{13058658062}{53887063}a^{10}+\frac{72005134705}{53887063}a^{9}+\frac{35505141340}{53887063}a^{8}-\frac{131684312811}{107774126}a^{7}-\frac{66512340079}{107774126}a^{6}+\frac{62493258927}{107774126}a^{5}+\frac{32432824785}{107774126}a^{4}-\frac{9100226232}{53887063}a^{3}-\frac{4915563287}{53887063}a^{2}+\frac{2019279199}{107774126}a+\frac{1199847919}{107774126}$, $\frac{136423970}{53887063}a^{20}+\frac{24874663}{53887063}a^{18}+\frac{803378}{53887063}a^{16}-\frac{6003195752}{53887063}a^{14}-\frac{2594585620}{53887063}a^{12}-\frac{12313237415}{53887063}a^{10}+\frac{33842300870}{53887063}a^{8}-\frac{31043956943}{53887063}a^{6}+\frac{14382353044}{53887063}a^{4}-\frac{4127266053}{53887063}a^{2}+\frac{524470293}{53887063}$, $\frac{105820985}{53887063}a^{21}+\frac{13179363}{53887063}a^{19}-\frac{6881422}{53887063}a^{17}-\frac{4664330621}{53887063}a^{15}-\frac{1750880419}{53887063}a^{13}-\frac{9160356101}{53887063}a^{11}+\frac{27210389195}{53887063}a^{9}-\frac{24617799207}{53887063}a^{7}+\frac{11902666602}{53887063}a^{5}-\frac{3226405466}{53887063}a^{3}+\frac{359412594}{53887063}a$, $\frac{151615684}{53887063}a^{21}+\frac{68985082}{53887063}a^{20}+\frac{37592885}{53887063}a^{19}+\frac{40304457}{53887063}a^{18}+\frac{8539222}{53887063}a^{17}+\frac{12722426}{53887063}a^{16}-\frac{6668284663}{53887063}a^{15}-\frac{3032621277}{53887063}a^{14}-\frac{3321029265}{53887063}a^{13}-\frac{2531176275}{53887063}a^{12}-\frac{14132127153}{53887063}a^{11}-\frac{7073077675}{53887063}a^{10}+\frac{36499993365}{53887063}a^{9}+\frac{14404182735}{53887063}a^{8}-\frac{32585986366}{53887063}a^{7}-\frac{9521104193}{53887063}a^{6}+\frac{15030217071}{53887063}a^{5}+\frac{2621030053}{53887063}a^{4}-\frac{4184662945}{53887063}a^{3}-\frac{448972856}{53887063}a^{2}+\frac{355828577}{53887063}a-\frac{112360946}{53887063}$, $\frac{68212505}{107774126}a^{21}-\frac{41191381}{107774126}a^{20}+\frac{4921093}{107774126}a^{19}-\frac{48855015}{107774126}a^{18}-\frac{7982665}{107774126}a^{17}-\frac{18723755}{107774126}a^{16}-\frac{3008073157}{107774126}a^{15}+\frac{1808431137}{107774126}a^{14}-\frac{485370436}{53887063}a^{13}+\frac{1301462776}{53887063}a^{12}-\frac{2851554806}{53887063}a^{11}+\frac{2495374581}{53887063}a^{10}+\frac{8979383372}{53887063}a^{9}-\frac{3102781668}{53887063}a^{8}-\frac{16532866837}{107774126}a^{7}+\frac{108025309}{107774126}a^{6}+\frac{7613855567}{107774126}a^{5}+\frac{2359615927}{107774126}a^{4}-\frac{1138684479}{53887063}a^{3}-\frac{459490382}{53887063}a^{2}+\frac{385549015}{107774126}a+\frac{279445799}{107774126}$, $\frac{12600711}{53887063}a^{21}+\frac{21381177}{53887063}a^{20}-\frac{3032345}{53887063}a^{19}+\frac{8992878}{53887063}a^{18}-\frac{6427766}{53887063}a^{17}+\frac{4407930}{53887063}a^{16}-\frac{561036068}{53887063}a^{15}-\frac{937546885}{53887063}a^{14}-\frac{9747549}{53887063}a^{13}-\frac{628311306}{53887063}a^{12}-\frac{795336334}{53887063}a^{11}-\frac{2171574686}{53887063}a^{10}+\frac{3955209384}{53887063}a^{9}+\frac{4664249104}{53887063}a^{8}-\frac{3426301487}{53887063}a^{7}-\frac{4043353982}{53887063}a^{6}+\frac{1821988689}{53887063}a^{5}+\frac{1567099503}{53887063}a^{4}-\frac{686390568}{53887063}a^{3}-\frac{450931773}{53887063}a^{2}+\frac{195001754}{53887063}a+\frac{58395410}{53887063}$, $\frac{188629917}{53887063}a^{20}+\frac{49981184}{53887063}a^{18}+\frac{13124170}{53887063}a^{16}-\frac{8295727888}{53887063}a^{14}-\frac{4272614354}{53887063}a^{12}-\frac{17727659853}{53887063}a^{10}+\frac{45080557815}{53887063}a^{8}-\frac{39937121030}{53887063}a^{6}+\frac{18412656726}{53887063}a^{4}-\frac{5324492156}{53887063}a^{2}+\frac{596625030}{53887063}$, $\frac{587195720}{53887063}a^{21}-\frac{275066481}{53887063}a^{20}+\frac{139672931}{53887063}a^{19}-\frac{74048265}{53887063}a^{18}+\frac{18671623}{53887063}a^{17}-\frac{19482133}{53887063}a^{16}-\frac{25839105081}{53887063}a^{15}+\frac{12097750228}{53887063}a^{14}-\frac{12607523127}{53887063}a^{13}+\frac{6283325105}{53887063}a^{12}-\frac{54031870358}{53887063}a^{11}+\frac{25878280883}{53887063}a^{10}+\frac{142636004423}{53887063}a^{9}-\frac{65658123016}{53887063}a^{8}-\frac{126102625022}{53887063}a^{7}+\frac{57893496485}{53887063}a^{6}+\frac{57224135687}{53887063}a^{5}-\frac{26721614846}{53887063}a^{4}-\frac{16426624099}{53887063}a^{3}+\frac{7715651939}{53887063}a^{2}+\frac{1662962038}{53887063}a-\frac{854818681}{53887063}$, $\frac{149700399}{107774126}a^{21}+\frac{13471187}{107774126}a^{20}+\frac{74792953}{107774126}a^{19}-\frac{248729}{107774126}a^{18}+\frac{5589347}{107774126}a^{17}+\frac{6999973}{107774126}a^{16}-\frac{6598525455}{107774126}a^{15}-\frac{588789235}{107774126}a^{14}-\frac{2472285921}{53887063}a^{13}-\frac{68637903}{53887063}a^{12}-\frac{7122993362}{53887063}a^{11}-\frac{745420647}{53887063}a^{10}+\frac{16696551649}{53887063}a^{9}+\frac{1662840470}{53887063}a^{8}-\frac{21470385167}{107774126}a^{7}-\frac{4424426193}{107774126}a^{6}+\frac{5217918787}{107774126}a^{5}+\frac{3668118697}{107774126}a^{4}-\frac{347066190}{53887063}a^{3}-\frac{788297234}{53887063}a^{2}-\frac{93399843}{107774126}a+\frac{150907333}{107774126}$
| 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: | \( 65776820.1374 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 65776820.1374 \cdot 1}{2\cdot\sqrt{1249711388122384634961076169998336}}\cr\approx \mathstrut & 0.356859029463 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 44*x^16 - 11*x^14 - 88*x^12 + 264*x^10 - 275*x^8 + 154*x^6 - 55*x^4 + 11*x^2 - 1)
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 - 44*x^16 - 11*x^14 - 88*x^12 + 264*x^10 - 275*x^8 + 154*x^6 - 55*x^4 + 11*x^2 - 1, 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 - 44*x^16 - 11*x^14 - 88*x^12 + 264*x^10 - 275*x^8 + 154*x^6 - 55*x^4 + 11*x^2 - 1);
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 - 44*x^16 - 11*x^14 - 88*x^12 + 264*x^10 - 275*x^8 + 154*x^6 - 55*x^4 + 11*x^2 - 1);
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
$C_2^{10}.F_{11}$ (as 22T34):
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 solvable group of order 112640 |
| The 44 conjugacy class representatives for $C_2^{10}.F_{11}$ |
| Character table for $C_2^{10}.F_{11}$ is not computed |
Intermediate fields
| 11.1.34522712143931.1 |
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 |
| Degree 44 siblings: | 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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/7.2.0.1}{2} }$ | R | $20{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.5.0.1}{5} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| Deg $20$ | $2$ | $10$ | $20$ | ||||
|
\(11\)
| 11.11.13.6 | $x^{11} + 44 x^{3} + 11$ | $11$ | $1$ | $13$ | $F_{11}$ | $[13/10]_{10}$ |
| 11.11.13.6 | $x^{11} + 44 x^{3} + 11$ | $11$ | $1$ | $13$ | $F_{11}$ | $[13/10]_{10}$ |