Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 26*x^20 + 246*x^18 + 992*x^16 + 1270*x^14 - 1382*x^12 - 3114*x^10 - 1149*x^8 + 264*x^6 + 117*x^4 - 5*x^2 - 1)
gp: K = bnfinit(y^22 + 26*y^20 + 246*y^18 + 992*y^16 + 1270*y^14 - 1382*y^12 - 3114*y^10 - 1149*y^8 + 264*y^6 + 117*y^4 - 5*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 26*x^20 + 246*x^18 + 992*x^16 + 1270*x^14 - 1382*x^12 - 3114*x^10 - 1149*x^8 + 264*x^6 + 117*x^4 - 5*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 26*x^20 + 246*x^18 + 992*x^16 + 1270*x^14 - 1382*x^12 - 3114*x^10 - 1149*x^8 + 264*x^6 + 117*x^4 - 5*x^2 - 1)
\( x^{22} + 26 x^{20} + 246 x^{18} + 992 x^{16} + 1270 x^{14} - 1382 x^{12} - 3114 x^{10} - 1149 x^{8} + \cdots - 1 \)
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: | $[6, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(56501459388151144478039723653407440896\)
\(\medspace = 2^{22}\cdot 1297^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(52.00\) | 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: |
\(2\), \(1297\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{676428174595}a^{20}-\frac{19771560887}{676428174595}a^{18}-\frac{162779306418}{676428174595}a^{16}-\frac{133181297699}{676428174595}a^{14}-\frac{152330887553}{676428174595}a^{12}-\frac{81694604668}{676428174595}a^{10}-\frac{53383332203}{135285634919}a^{8}-\frac{320206232764}{676428174595}a^{6}+\frac{139429616591}{676428174595}a^{4}-\frac{141564139586}{676428174595}a^{2}-\frac{106712717787}{676428174595}$, $\frac{1}{676428174595}a^{21}-\frac{19771560887}{676428174595}a^{19}-\frac{162779306418}{676428174595}a^{17}-\frac{133181297699}{676428174595}a^{15}-\frac{152330887553}{676428174595}a^{13}-\frac{81694604668}{676428174595}a^{11}-\frac{53383332203}{135285634919}a^{9}-\frac{320206232764}{676428174595}a^{7}+\frac{139429616591}{676428174595}a^{5}-\frac{141564139586}{676428174595}a^{3}-\frac{106712717787}{676428174595}a$
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: |
$a$, $\frac{855255824341}{676428174595}a^{21}+\frac{21817612599078}{676428174595}a^{19}+\frac{199831743137357}{676428174595}a^{17}+\frac{753750350975166}{676428174595}a^{15}+\frac{746072548268822}{676428174595}a^{13}-\frac{14\!\cdots\!78}{676428174595}a^{11}-\frac{374267195273595}{135285634919}a^{9}-\frac{302892576568599}{676428174595}a^{7}+\frac{153734075746076}{676428174595}a^{5}+\frac{31798902500804}{676428174595}a^{3}-\frac{783146458952}{676428174595}a$, $\frac{212229776901}{676428174595}a^{21}+\frac{5353165898548}{676428174595}a^{19}+\frac{48034478850727}{676428174595}a^{17}+\frac{172787526142636}{676428174595}a^{15}+\frac{131152096297532}{676428174595}a^{13}-\frac{412162021955058}{676428174595}a^{11}-\frac{72580777624767}{135285634919}a^{9}+\frac{61170841035316}{676428174595}a^{7}+\frac{62715924086566}{676428174595}a^{5}-\frac{3903182387741}{676428174595}a^{3}-\frac{3806556022887}{676428174595}a$, $\frac{11444575079}{676428174595}a^{20}+\frac{303332846647}{676428174595}a^{18}+\frac{2958672232698}{676428174595}a^{16}+\frac{12604786931164}{676428174595}a^{14}+\frac{18801781535013}{676428174595}a^{12}-\frac{13309670642767}{676428174595}a^{10}-\frac{9063860424726}{135285634919}a^{8}-\frac{16265282915526}{676428174595}a^{6}+\frac{2317504951539}{676428174595}a^{4}-\frac{1935131597089}{676428174595}a^{2}-\frac{426411426593}{676428174595}$, $\frac{665898757678}{676428174595}a^{21}+\frac{17272488016704}{676428174595}a^{19}+\frac{162844722387701}{676428174595}a^{17}+\frac{652952954406193}{676428174595}a^{15}+\frac{827085133714646}{676428174595}a^{13}-\frac{892498136994669}{676428174595}a^{11}-\frac{390387911433412}{135285634919}a^{9}-\frac{814053891765917}{676428174595}a^{7}+\frac{56833190799858}{676428174595}a^{5}+\frac{67592514079707}{676428174595}a^{3}+\frac{5156763927704}{676428174595}a$, $\frac{307755205118}{676428174595}a^{20}+\frac{7754952460429}{676428174595}a^{18}+\frac{69471421784741}{676428174595}a^{16}+\frac{249085212641238}{676428174595}a^{14}+\frac{186343343323606}{676428174595}a^{12}-\frac{593461018510004}{676428174595}a^{10}-\frac{100662735641095}{135285634919}a^{8}+\frac{82332436742648}{676428174595}a^{6}+\frac{66778338019053}{676428174595}a^{4}-\frac{6012763251453}{676428174595}a^{2}-\frac{453308446736}{676428174595}$, $\frac{29222134253}{676428174595}a^{20}+\frac{827054233349}{676428174595}a^{18}+\frac{8879208937451}{676428174595}a^{16}+\frac{44059034050013}{676428174595}a^{14}+\frac{90575889992326}{676428174595}a^{12}-\frac{2723505600149}{676428174595}a^{10}-\frac{44160488854231}{135285634919}a^{8}-\frac{132517507086382}{676428174595}a^{6}+\frac{26169209088213}{676428174595}a^{4}+\frac{11899496143307}{676428174595}a^{2}-\frac{1737282995531}{676428174595}$, $\frac{588749916646}{676428174595}a^{20}+\frac{14825821133553}{676428174595}a^{18}+\frac{132695058449772}{676428174595}a^{16}+\frac{475285338702666}{676428174595}a^{14}+\frac{357259084305722}{676428174595}a^{12}-\frac{11\!\cdots\!98}{676428174595}a^{10}-\frac{184664253958848}{135285634919}a^{8}+\frac{100206893186836}{676428174595}a^{6}+\frac{74976954776406}{676428174595}a^{4}-\frac{8579094359001}{676428174595}a^{2}-\frac{594754063312}{676428174595}$, $\frac{60106933698}{135285634919}a^{21}+\frac{1536643599219}{135285634919}a^{19}+\frac{14135024326215}{135285634919}a^{17}+\frac{53910481334377}{135285634919}a^{15}+\frac{56832322471451}{135285634919}a^{13}-\frac{92977456547940}{135285634919}a^{11}-\frac{132281057241590}{135285634919}a^{9}-\frac{39919194634413}{135285634919}a^{7}-\frac{2329679465746}{135285634919}a^{5}+\frac{2250197347027}{135285634919}a^{3}+\frac{775723175083}{135285634919}a$, $\frac{604535290122}{676428174595}a^{21}+\frac{15229215489121}{676428174595}a^{19}+\frac{136385123670399}{676428174595}a^{17}+\frac{488946870730052}{676428174595}a^{15}+\frac{367838885875919}{676428174595}a^{13}-\frac{11\!\cdots\!46}{676428174595}a^{11}-\frac{194677030663554}{135285634919}a^{9}+\frac{120893423574687}{676428174595}a^{7}+\frac{114257695506412}{676428174595}a^{5}-\frac{2946121807977}{676428174595}a^{3}-\frac{3920911053334}{676428174595}a$, $\frac{9967929987126}{676428174595}a^{21}+\frac{11137496451888}{676428174595}a^{20}+\frac{247863229744943}{676428174595}a^{19}+\frac{297361844903334}{676428174595}a^{18}+\frac{21\!\cdots\!17}{676428174595}a^{17}+\frac{29\!\cdots\!26}{676428174595}a^{16}+\frac{69\!\cdots\!61}{676428174595}a^{15}+\frac{13\!\cdots\!73}{676428174595}a^{14}+\frac{258784791066877}{676428174595}a^{13}+\frac{22\!\cdots\!56}{676428174595}a^{12}-\frac{33\!\cdots\!93}{676428174595}a^{11}-\frac{26\!\cdots\!59}{676428174595}a^{10}-\frac{45\!\cdots\!75}{135285634919}a^{9}-\frac{82\!\cdots\!92}{135285634919}a^{8}+\frac{27\!\cdots\!26}{676428174595}a^{7}-\frac{39\!\cdots\!72}{676428174595}a^{6}+\frac{31\!\cdots\!36}{676428174595}a^{5}-\frac{14\!\cdots\!97}{676428174595}a^{4}+\frac{84\!\cdots\!34}{676428174595}a^{3}-\frac{23\!\cdots\!08}{676428174595}a^{2}+\frac{499586131220883}{676428174595}a-\frac{113487896774871}{676428174595}$, $\frac{616618884954}{135285634919}a^{21}-\frac{267873765887}{676428174595}a^{20}+\frac{15964429734210}{135285634919}a^{19}-\frac{6315691800146}{676428174595}a^{18}+\frac{149713972399191}{135285634919}a^{17}-\frac{47650979784589}{676428174595}a^{16}+\frac{589166334874755}{135285634919}a^{15}-\frac{68832620598772}{676428174595}a^{14}+\frac{655325884337178}{135285634919}a^{13}+\frac{684511304823811}{676428174595}a^{12}-\frac{12\!\cdots\!50}{135285634919}a^{11}+\frac{30\!\cdots\!16}{676428174595}a^{10}-\frac{24\!\cdots\!77}{135285634919}a^{9}+\frac{782037433130591}{135285634919}a^{8}-\frac{11\!\cdots\!80}{135285634919}a^{7}+\frac{15\!\cdots\!43}{676428174595}a^{6}+\frac{78951513284563}{135285634919}a^{5}-\frac{156603460146902}{676428174595}a^{4}+\frac{93390725833290}{135285634919}a^{3}-\frac{150240366334018}{676428174595}a^{2}+\frac{6419910346873}{135285634919}a-\frac{10507514306011}{676428174595}$, $\frac{24\!\cdots\!82}{676428174595}a^{21}-\frac{16\!\cdots\!17}{135285634919}a^{20}+\frac{64\!\cdots\!21}{676428174595}a^{19}-\frac{42\!\cdots\!61}{135285634919}a^{18}+\frac{61\!\cdots\!19}{676428174595}a^{17}-\frac{40\!\cdots\!16}{135285634919}a^{16}+\frac{25\!\cdots\!62}{676428174595}a^{15}-\frac{16\!\cdots\!09}{135285634919}a^{14}+\frac{34\!\cdots\!79}{676428174595}a^{13}-\frac{22\!\cdots\!93}{135285634919}a^{12}-\frac{30\!\cdots\!66}{676428174595}a^{11}+\frac{20\!\cdots\!83}{135285634919}a^{10}-\frac{16\!\cdots\!75}{135285634919}a^{9}+\frac{53\!\cdots\!29}{135285634919}a^{8}-\frac{37\!\cdots\!13}{676428174595}a^{7}+\frac{24\!\cdots\!69}{135285634919}a^{6}+\frac{24\!\cdots\!67}{676428174595}a^{5}-\frac{16\!\cdots\!79}{135285634919}a^{4}+\frac{31\!\cdots\!13}{676428174595}a^{3}-\frac{21\!\cdots\!84}{135285634919}a^{2}+\frac{22\!\cdots\!56}{676428174595}a-\frac{14\!\cdots\!23}{135285634919}$
| 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: | \( 32131005113.8 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot 32131005113.8 \cdot 1}{2\cdot\sqrt{56501459388151144478039723653407440896}}\cr\approx \mathstrut & 0.332264087708 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 26*x^20 + 246*x^18 + 992*x^16 + 1270*x^14 - 1382*x^12 - 3114*x^10 - 1149*x^8 + 264*x^6 + 117*x^4 - 5*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 + 26*x^20 + 246*x^18 + 992*x^16 + 1270*x^14 - 1382*x^12 - 3114*x^10 - 1149*x^8 + 264*x^6 + 117*x^4 - 5*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 + 26*x^20 + 246*x^18 + 992*x^16 + 1270*x^14 - 1382*x^12 - 3114*x^10 - 1149*x^8 + 264*x^6 + 117*x^4 - 5*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 + 26*x^20 + 246*x^18 + 992*x^16 + 1270*x^14 - 1382*x^12 - 3114*x^10 - 1149*x^8 + 264*x^6 + 117*x^4 - 5*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
$C_2^{10}.D_{11}$ (as 22T30):
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 22528 |
| The 100 conjugacy class representatives for $C_2^{10}.D_{11}$ |
| Character table for $C_2^{10}.D_{11}$ |
Intermediate fields
| 11.11.3670285774226257.1 |
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 siblings: | data not computed |
| Degree 44 siblings: | data not computed |
| Minimal sibling: | 22.10.73282392826432034388017521578469450842112.6 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.11.0.1}{11} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{5}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{5}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 $22$ | $2$ | $11$ | $22$ | |||
|
\(1297\)
| $\Q_{1297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |