Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 44*x^10 + 44*x^9 + 847*x^8 - 924*x^7 - 8844*x^6 + 19272*x^5 + 20075*x^4 - 156442*x^3 + 83600*x^2 + 359236*x - 470323)
gp: K = bnfinit(y^12 - 2*y^11 - 44*y^10 + 44*y^9 + 847*y^8 - 924*y^7 - 8844*y^6 + 19272*y^5 + 20075*y^4 - 156442*y^3 + 83600*y^2 + 359236*y - 470323, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 - 44*x^10 + 44*x^9 + 847*x^8 - 924*x^7 - 8844*x^6 + 19272*x^5 + 20075*x^4 - 156442*x^3 + 83600*x^2 + 359236*x - 470323);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 - 44*x^10 + 44*x^9 + 847*x^8 - 924*x^7 - 8844*x^6 + 19272*x^5 + 20075*x^4 - 156442*x^3 + 83600*x^2 + 359236*x - 470323)
\( x^{12} - 2 x^{11} - 44 x^{10} + 44 x^{9} + 847 x^{8} - 924 x^{7} - 8844 x^{6} + 19272 x^{5} + \cdots - 470323 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-10069154974041885785533184\)
\(\medspace = -\,2^{8}\cdot 11^{11}\cdot 13^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(121.22\) | 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: | $2^{2/3}11^{119/110}13^{10/11}\approx 218.75600312740116$ | ||
| Ramified primes: |
\(2\), \(11\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{24}a^{8}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{7}{24}$, $\frac{1}{48}a^{9}-\frac{1}{48}a^{8}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{8}a^{3}-\frac{1}{8}a^{2}+\frac{11}{48}a+\frac{13}{48}$, $\frac{1}{144}a^{10}-\frac{1}{144}a^{8}-\frac{1}{12}a^{7}+\frac{1}{24}a^{6}+\frac{1}{12}a^{5}+\frac{1}{24}a^{4}-\frac{1}{4}a^{3}+\frac{65}{144}a^{2}+\frac{1}{4}a+\frac{67}{144}$, $\frac{1}{82\!\cdots\!76}a^{11}+\frac{55\!\cdots\!63}{74\!\cdots\!16}a^{10}-\frac{19\!\cdots\!61}{37\!\cdots\!08}a^{9}-\frac{40\!\cdots\!77}{37\!\cdots\!08}a^{8}-\frac{43\!\cdots\!53}{51\!\cdots\!89}a^{7}+\frac{10\!\cdots\!43}{62\!\cdots\!68}a^{6}+\frac{15\!\cdots\!43}{12\!\cdots\!36}a^{5}+\frac{64\!\cdots\!15}{12\!\cdots\!36}a^{4}+\frac{12\!\cdots\!77}{74\!\cdots\!16}a^{3}-\frac{13\!\cdots\!43}{74\!\cdots\!16}a^{2}-\frac{60\!\cdots\!83}{18\!\cdots\!04}a+\frac{14\!\cdots\!31}{20\!\cdots\!44}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: | $6$ | 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: |
$\frac{1025169327805}{22\!\cdots\!16}a^{11}-\frac{16\!\cdots\!51}{74\!\cdots\!16}a^{10}-\frac{136576463254414}{15\!\cdots\!67}a^{9}+\frac{91\!\cdots\!45}{74\!\cdots\!16}a^{8}+\frac{59\!\cdots\!20}{15\!\cdots\!67}a^{7}-\frac{23\!\cdots\!83}{12\!\cdots\!36}a^{6}-\frac{40\!\cdots\!69}{62\!\cdots\!68}a^{5}+\frac{26\!\cdots\!61}{12\!\cdots\!36}a^{4}+\frac{52\!\cdots\!31}{20\!\cdots\!56}a^{3}-\frac{82\!\cdots\!03}{74\!\cdots\!16}a^{2}+\frac{12\!\cdots\!11}{62\!\cdots\!68}a+\frac{57\!\cdots\!71}{82\!\cdots\!76}$, $\frac{2841978155909}{13\!\cdots\!62}a^{11}-\frac{7164756392923}{17\!\cdots\!16}a^{10}-\frac{11\!\cdots\!27}{10\!\cdots\!96}a^{9}+\frac{287379894068303}{35\!\cdots\!32}a^{8}+\frac{41\!\cdots\!35}{17\!\cdots\!16}a^{7}-\frac{13\!\cdots\!49}{17\!\cdots\!16}a^{6}-\frac{12\!\cdots\!95}{43\!\cdots\!54}a^{5}+\frac{12\!\cdots\!83}{731393724896759}a^{4}+\frac{66\!\cdots\!53}{52\!\cdots\!48}a^{3}-\frac{13\!\cdots\!19}{87\!\cdots\!08}a^{2}-\frac{34\!\cdots\!97}{10\!\cdots\!96}a+\frac{95\!\cdots\!35}{11\!\cdots\!44}$, $\frac{46\!\cdots\!77}{41\!\cdots\!88}a^{11}-\frac{26\!\cdots\!93}{37\!\cdots\!08}a^{10}-\frac{14\!\cdots\!05}{74\!\cdots\!16}a^{9}+\frac{98\!\cdots\!33}{74\!\cdots\!16}a^{8}+\frac{16\!\cdots\!27}{41\!\cdots\!12}a^{7}-\frac{33\!\cdots\!37}{12\!\cdots\!36}a^{6}+\frac{26\!\cdots\!61}{15\!\cdots\!67}a^{5}+\frac{90\!\cdots\!29}{62\!\cdots\!68}a^{4}-\frac{18\!\cdots\!76}{46\!\cdots\!01}a^{3}-\frac{12\!\cdots\!63}{18\!\cdots\!04}a^{2}+\frac{92\!\cdots\!41}{74\!\cdots\!16}a-\frac{10\!\cdots\!67}{82\!\cdots\!76}$, $\frac{42\!\cdots\!87}{82\!\cdots\!76}a^{11}-\frac{80\!\cdots\!77}{24\!\cdots\!72}a^{10}-\frac{65\!\cdots\!89}{74\!\cdots\!16}a^{9}+\frac{15\!\cdots\!93}{24\!\cdots\!72}a^{8}+\frac{22\!\cdots\!93}{12\!\cdots\!36}a^{7}-\frac{15\!\cdots\!15}{12\!\cdots\!36}a^{6}+\frac{95\!\cdots\!25}{12\!\cdots\!36}a^{5}+\frac{27\!\cdots\!63}{41\!\cdots\!12}a^{4}-\frac{13\!\cdots\!11}{74\!\cdots\!16}a^{3}-\frac{77\!\cdots\!33}{24\!\cdots\!72}a^{2}+\frac{41\!\cdots\!35}{74\!\cdots\!16}a-\frac{51\!\cdots\!51}{91\!\cdots\!64}$, $\frac{74\!\cdots\!13}{51\!\cdots\!11}a^{11}-\frac{30\!\cdots\!79}{20\!\cdots\!56}a^{10}-\frac{30\!\cdots\!24}{46\!\cdots\!01}a^{9}-\frac{25\!\cdots\!37}{12\!\cdots\!36}a^{8}+\frac{19\!\cdots\!24}{15\!\cdots\!67}a^{7}-\frac{39\!\cdots\!37}{31\!\cdots\!34}a^{6}-\frac{42\!\cdots\!87}{31\!\cdots\!34}a^{5}+\frac{27\!\cdots\!51}{20\!\cdots\!56}a^{4}+\frac{22\!\cdots\!88}{46\!\cdots\!01}a^{3}-\frac{36\!\cdots\!65}{20\!\cdots\!56}a^{2}-\frac{71\!\cdots\!19}{93\!\cdots\!02}a+\frac{23\!\cdots\!19}{45\!\cdots\!32}$, $\frac{36\!\cdots\!63}{37\!\cdots\!08}a^{11}-\frac{14\!\cdots\!33}{74\!\cdots\!16}a^{10}-\frac{17\!\cdots\!01}{37\!\cdots\!08}a^{9}+\frac{31\!\cdots\!51}{74\!\cdots\!16}a^{8}+\frac{22\!\cdots\!13}{20\!\cdots\!56}a^{7}-\frac{66\!\cdots\!77}{12\!\cdots\!36}a^{6}-\frac{42\!\cdots\!97}{31\!\cdots\!34}a^{5}+\frac{11\!\cdots\!99}{12\!\cdots\!36}a^{4}+\frac{28\!\cdots\!15}{37\!\cdots\!08}a^{3}-\frac{53\!\cdots\!45}{74\!\cdots\!16}a^{2}-\frac{54\!\cdots\!91}{37\!\cdots\!08}a+\frac{14\!\cdots\!35}{74\!\cdots\!16}$
| 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: | \( 826076900.482 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 826076900.482 \cdot 1}{2\cdot\sqrt{10069154974041885785533184}}\cr\approx \mathstrut & 5.09862772338 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 44*x^10 + 44*x^9 + 847*x^8 - 924*x^7 - 8844*x^6 + 19272*x^5 + 20075*x^4 - 156442*x^3 + 83600*x^2 + 359236*x - 470323)
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^12 - 2*x^11 - 44*x^10 + 44*x^9 + 847*x^8 - 924*x^7 - 8844*x^6 + 19272*x^5 + 20075*x^4 - 156442*x^3 + 83600*x^2 + 359236*x - 470323, 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^12 - 2*x^11 - 44*x^10 + 44*x^9 + 847*x^8 - 924*x^7 - 8844*x^6 + 19272*x^5 + 20075*x^4 - 156442*x^3 + 83600*x^2 + 359236*x - 470323);
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^12 - 2*x^11 - 44*x^10 + 44*x^9 + 847*x^8 - 924*x^7 - 8844*x^6 + 19272*x^5 + 20075*x^4 - 156442*x^3 + 83600*x^2 + 359236*x - 470323);
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
$\PGL(2,11)$ (as 12T218):
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 1320 |
| The 13 conjugacy class representatives for $\PGL(2,11)$ |
| Character table for $\PGL(2,11)$ |
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]
Sibling fields
| Degree 22 sibling: | data not computed |
| Degree 24 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 | ${\href{/padicField/3.2.0.1}{2} }^{6}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(11\)
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 11.11.11.10 | $x^{11} + 33 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 13.11.10.1 | $x^{11} + 13$ | $11$ | $1$ | $10$ | $F_{11}$ | $[\ ]_{11}^{10}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 10.251...296.55.a.a | $10$ | $ 2^{6} \cdot 11^{11} \cdot 13^{10}$ | 12.2.10069154974041885785533184.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
| 10.100...184.55.a.a | $10$ | $ 2^{8} \cdot 11^{11} \cdot 13^{10}$ | 12.2.10069154974041885785533184.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
| 10.251...296.110.a.a | $10$ | $ 2^{6} \cdot 11^{11} \cdot 13^{10}$ | 12.2.10069154974041885785533184.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
| 10.251...296.22t14.a.a | $10$ | $ 2^{6} \cdot 11^{11} \cdot 13^{10}$ | 12.2.10069154974041885785533184.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
| 10.251...296.22t14.a.b | $10$ | $ 2^{6} \cdot 11^{11} \cdot 13^{10}$ | 12.2.10069154974041885785533184.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
| * | 11.100...184.12t218.a.a | $11$ | $ 2^{8} \cdot 11^{11} \cdot 13^{10}$ | 12.2.10069154974041885785533184.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $1$ |
| 11.110...024.24t2949.a.a | $11$ | $ 2^{8} \cdot 11^{12} \cdot 13^{10}$ | 12.2.10069154974041885785533184.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $-1$ | |
| 12.121...264.55.a.a | $12$ | $ 2^{8} \cdot 11^{13} \cdot 13^{10}$ | 12.2.10069154974041885785533184.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $2$ | |
| 12.121...264.110.a.a | $12$ | $ 2^{8} \cdot 11^{13} \cdot 13^{10}$ | 12.2.10069154974041885785533184.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $-2$ | |
| 12.121...264.110.a.b | $12$ | $ 2^{8} \cdot 11^{13} \cdot 13^{10}$ | 12.2.10069154974041885785533184.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $-2$ | |
| 12.121...264.55.a.b | $12$ | $ 2^{8} \cdot 11^{13} \cdot 13^{10}$ | 12.2.10069154974041885785533184.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $2$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.