Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 22*x^10 - 55*x^9 - 165*x^8 - 132*x^7 + 55*x^6 - 66*x^5 - 495*x^4 - 385*x^3 - 319*x^2 - 55*x - 1331)
gp: K = bnfinit(y^12 - 22*y^10 - 55*y^9 - 165*y^8 - 132*y^7 + 55*y^6 - 66*y^5 - 495*y^4 - 385*y^3 - 319*y^2 - 55*y - 1331, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 22*x^10 - 55*x^9 - 165*x^8 - 132*x^7 + 55*x^6 - 66*x^5 - 495*x^4 - 385*x^3 - 319*x^2 - 55*x - 1331);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 22*x^10 - 55*x^9 - 165*x^8 - 132*x^7 + 55*x^6 - 66*x^5 - 495*x^4 - 385*x^3 - 319*x^2 - 55*x - 1331)
\( x^{12} - 22 x^{10} - 55 x^{9} - 165 x^{8} - 132 x^{7} + 55 x^{6} - 66 x^{5} - 495 x^{4} - 385 x^{3} + \cdots - 1331 \)
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: |
\(-1371945240568483487545135739\)
\(\medspace = -\,11^{11}\cdot 37^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(182.58\) | 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: | $11^{119/110}37^{10/11}\approx 356.64445176161007$ | ||
| Ramified primes: |
\(11\), \(37\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{6}a^{6}+\frac{1}{6}a^{4}+\frac{1}{6}a^{3}+\frac{1}{6}a^{2}-\frac{1}{6}a-\frac{1}{6}$, $\frac{1}{222}a^{7}+\frac{2}{37}a^{6}-\frac{17}{222}a^{5}-\frac{16}{111}a^{4}+\frac{43}{222}a^{3}+\frac{35}{222}a^{2}+\frac{31}{111}a-\frac{29}{74}$, $\frac{1}{444}a^{8}+\frac{2}{37}a^{6}-\frac{25}{222}a^{5}-\frac{9}{74}a^{4}+\frac{1}{3}a^{3}+\frac{49}{444}a^{2}+\frac{47}{222}a+\frac{193}{444}$, $\frac{1}{444}a^{9}+\frac{8}{111}a^{6}-\frac{15}{74}a^{5}-\frac{23}{222}a^{4}-\frac{169}{444}a^{3}+\frac{17}{111}a^{2}+\frac{1}{4}a-\frac{29}{222}$, $\frac{1}{444}a^{10}-\frac{5}{74}a^{6}+\frac{9}{74}a^{5}-\frac{11}{148}a^{4}+\frac{2}{37}a^{3}-\frac{121}{444}a^{2}+\frac{89}{222}a+\frac{10}{37}$, $\frac{1}{4884}a^{11}+\frac{1}{111}a^{6}-\frac{3}{148}a^{5}-\frac{19}{222}a^{4}+\frac{113}{444}a^{3}+\frac{13}{74}a^{2}-\frac{95}{222}a+\frac{50}{111}$
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: | $6$ | 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: |
$\frac{1253}{444}a^{11}+\frac{229}{111}a^{10}-\frac{9375}{148}a^{9}-\frac{45385}{222}a^{8}-\frac{20495}{37}a^{7}-\frac{61856}{111}a^{6}+\frac{160729}{444}a^{5}+\frac{91369}{111}a^{4}-\frac{33655}{37}a^{3}-\frac{192089}{74}a^{2}-\frac{967343}{444}a+\frac{36953}{37}$, $\frac{1}{37}a^{11}+\frac{115}{222}a^{10}-\frac{319}{111}a^{9}-\frac{1151}{222}a^{8}-\frac{1889}{111}a^{7}+\frac{115}{111}a^{6}+\frac{6881}{74}a^{5}+\frac{42259}{222}a^{4}+\frac{21046}{111}a^{3}+\frac{16125}{74}a^{2}+\frac{16409}{74}a+\frac{22217}{74}$, $\frac{1233}{74}a^{11}-\frac{11965}{444}a^{10}-\frac{36439}{111}a^{9}-\frac{80249}{222}a^{8}-\frac{461669}{222}a^{7}+\frac{111499}{111}a^{6}-\frac{123541}{222}a^{5}-\frac{488361}{148}a^{4}-\frac{773381}{222}a^{3}-\frac{2377573}{444}a^{2}+\frac{899857}{222}a-\frac{500999}{37}$, $\frac{182741}{1628}a^{11}-\frac{11659}{111}a^{10}-\frac{112651}{37}a^{9}-\frac{1224503}{444}a^{8}+\frac{178375}{74}a^{7}+\frac{241823}{222}a^{6}-\frac{3645545}{444}a^{5}-\frac{1444657}{222}a^{4}-\frac{23185}{12}a^{3}+\frac{479643}{148}a^{2}-\frac{2861578}{111}a+\frac{331127}{148}$, $\frac{20201}{4884}a^{11}-\frac{618}{37}a^{10}-\frac{3477}{148}a^{9}-\frac{58903}{444}a^{8}-\frac{10867}{74}a^{7}+\frac{5120}{111}a^{6}+\frac{17609}{444}a^{5}-\frac{96107}{222}a^{4}-\frac{21919}{74}a^{3}-\frac{176287}{444}a^{2}+\frac{41345}{148}a-\frac{201625}{148}$, $\frac{121373}{111}a^{11}+\frac{9934}{111}a^{10}-\frac{303587}{12}a^{9}-\frac{14102225}{222}a^{8}-\frac{17535368}{111}a^{7}-\frac{2141512}{37}a^{6}+\frac{12308230}{37}a^{5}+\frac{32289700}{111}a^{4}-\frac{253032941}{444}a^{3}-\frac{94040272}{111}a^{2}-\frac{61736093}{444}a+\frac{86424803}{74}$
| 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: | \( 605851849.746 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 605851849.746 \cdot 1}{2\cdot\sqrt{1371945240568483487545135739}}\cr\approx \mathstrut & 0.320351961083 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 22*x^10 - 55*x^9 - 165*x^8 - 132*x^7 + 55*x^6 - 66*x^5 - 495*x^4 - 385*x^3 - 319*x^2 - 55*x - 1331)
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 - 22*x^10 - 55*x^9 - 165*x^8 - 132*x^7 + 55*x^6 - 66*x^5 - 495*x^4 - 385*x^3 - 319*x^2 - 55*x - 1331, 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 - 22*x^10 - 55*x^9 - 165*x^8 - 132*x^7 + 55*x^6 - 66*x^5 - 495*x^4 - 385*x^3 - 319*x^2 - 55*x - 1331);
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 - 22*x^10 - 55*x^9 - 165*x^8 - 132*x^7 + 55*x^6 - 66*x^5 - 495*x^4 - 385*x^3 - 319*x^2 - 55*x - 1331);
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 | ${\href{/padicField/2.4.0.1}{4} }^{3}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }$ | R | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{5}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 11.11.11.7 | $x^{11} + 44 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ | |
|
\(37\)
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 37.11.10.1 | $x^{11} + 37$ | $11$ | $1$ | $10$ | $C_{11}:C_5$ | $[\ ]_{11}^{5}$ |
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) | $1$ | $-1$ | |
| 10.137...739.55.a.a | $10$ | $ 11^{11} \cdot 37^{10}$ | 12.2.1371945240568483487545135739.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
| 10.137...739.55.b.a | $10$ | $ 11^{11} \cdot 37^{10}$ | 12.2.1371945240568483487545135739.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
| 10.137...739.110.a.a | $10$ | $ 11^{11} \cdot 37^{10}$ | 12.2.1371945240568483487545135739.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
| 10.137...739.22t14.a.a | $10$ | $ 11^{11} \cdot 37^{10}$ | 12.2.1371945240568483487545135739.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
| 10.137...739.22t14.a.b | $10$ | $ 11^{11} \cdot 37^{10}$ | 12.2.1371945240568483487545135739.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $0$ | |
| * | 11.137...739.12t218.a.a | $11$ | $ 11^{11} \cdot 37^{10}$ | 12.2.1371945240568483487545135739.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $1$ |
| 11.150...129.24t2949.a.a | $11$ | $ 11^{12} \cdot 37^{10}$ | 12.2.1371945240568483487545135739.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $-1$ | |
| 12.166...419.55.a.a | $12$ | $ 11^{13} \cdot 37^{10}$ | 12.2.1371945240568483487545135739.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $2$ | |
| 12.166...419.110.a.a | $12$ | $ 11^{13} \cdot 37^{10}$ | 12.2.1371945240568483487545135739.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $-2$ | |
| 12.166...419.110.a.b | $12$ | $ 11^{13} \cdot 37^{10}$ | 12.2.1371945240568483487545135739.1 | $\PGL(2,11)$ (as 12T218) | $1$ | $-2$ | |
| 12.166...419.55.a.b | $12$ | $ 11^{13} \cdot 37^{10}$ | 12.2.1371945240568483487545135739.1 | $\PGL(2,11)$ (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 *.