Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 27*x^10 - 10*x^9 + 129*x^8 - 987*x^7 + 3068*x^6 - 13047*x^5 + 24529*x^4 - 70220*x^3 + 105861*x^2 - 72135*x + 17885)
gp: K = bnfinit(y^12 - y^11 + 27*y^10 - 10*y^9 + 129*y^8 - 987*y^7 + 3068*y^6 - 13047*y^5 + 24529*y^4 - 70220*y^3 + 105861*y^2 - 72135*y + 17885, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 27*x^10 - 10*x^9 + 129*x^8 - 987*x^7 + 3068*x^6 - 13047*x^5 + 24529*x^4 - 70220*x^3 + 105861*x^2 - 72135*x + 17885);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 + 27*x^10 - 10*x^9 + 129*x^8 - 987*x^7 + 3068*x^6 - 13047*x^5 + 24529*x^4 - 70220*x^3 + 105861*x^2 - 72135*x + 17885)
\( x^{12} - x^{11} + 27 x^{10} - 10 x^{9} + 129 x^{8} - 987 x^{7} + 3068 x^{6} - 13047 x^{5} + 24529 x^{4} + \cdots + 17885 \)
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: |
\(-7972425761140316000000\)
\(\medspace = -\,2^{8}\cdot 5^{6}\cdot 7^{5}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(66.85\) | 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: | $2^{2/3}5^{1/2}7^{1/2}17^{3/4}\approx 78.62440592271739$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-119}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}+\frac{2}{7}a^{5}-\frac{3}{7}a^{4}+\frac{3}{7}a^{3}-\frac{3}{7}a^{2}$, $\frac{1}{35}a^{7}+\frac{2}{35}a^{6}+\frac{4}{35}a^{5}+\frac{17}{35}a^{4}-\frac{2}{7}a^{3}-\frac{2}{5}a^{2}$, $\frac{1}{175}a^{8}-\frac{1}{35}a^{6}-\frac{1}{175}a^{5}-\frac{29}{175}a^{4}-\frac{79}{175}a^{3}-\frac{62}{175}a^{2}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{350}a^{9}+\frac{9}{350}a^{6}+\frac{83}{175}a^{5}+\frac{3}{175}a^{4}+\frac{9}{50}a^{3}-\frac{2}{5}a^{2}-\frac{1}{5}a-\frac{1}{2}$, $\frac{1}{46550}a^{10}-\frac{6}{23275}a^{9}-\frac{24}{23275}a^{8}+\frac{11}{2450}a^{7}+\frac{824}{23275}a^{6}+\frac{7381}{23275}a^{5}+\frac{5083}{46550}a^{4}-\frac{10902}{23275}a^{3}+\frac{1374}{3325}a^{2}+\frac{59}{266}a+\frac{34}{95}$, $\frac{1}{157097957676100}a^{11}-\frac{70643429}{78548978838050}a^{10}-\frac{88461860943}{157097957676100}a^{9}-\frac{202427339977}{157097957676100}a^{8}-\frac{283340387993}{78548978838050}a^{7}-\frac{9525849700889}{157097957676100}a^{6}+\frac{48186586227243}{157097957676100}a^{5}+\frac{5107847946263}{11221282691150}a^{4}-\frac{2691398885073}{31419591535220}a^{3}-\frac{3787771192599}{22442565382300}a^{2}-\frac{622686379221}{2244256538230}a+\frac{139727417691}{641216153780}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $5$ |
Class group and class number
$C_{4}$, which has order $4$ (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: |
$\frac{16957526767}{78548978838050}a^{11}+\frac{85907513}{15709795767610}a^{10}+\frac{64930333751}{11221282691150}a^{9}+\frac{290311960979}{78548978838050}a^{8}+\frac{48790877753}{1603040384450}a^{7}-\frac{2901502891479}{15709795767610}a^{6}+\frac{36673177629657}{78548978838050}a^{5}-\frac{180975869069867}{78548978838050}a^{4}+\frac{228603036102697}{78548978838050}a^{3}-\frac{134186765774311}{11221282691150}a^{2}+\frac{24501581382989}{2244256538230}a-\frac{775001096351}{320608076890}$, $\frac{121503}{98847700}a^{11}-\frac{1854}{3530275}a^{10}+\frac{3238181}{98847700}a^{9}+\frac{657953}{98847700}a^{8}+\frac{785779}{4942385}a^{7}-\frac{3165947}{2824220}a^{6}+\frac{307710597}{98847700}a^{5}-\frac{349793991}{24711925}a^{4}+\frac{2126067019}{98847700}a^{3}-\frac{1024016121}{14121100}a^{2}+\frac{61110019}{706055}a-\frac{11548601}{403460}$, $\frac{26806789}{118118765170}a^{11}-\frac{470942604}{2067078390475}a^{10}+\frac{4672913593}{826831356190}a^{9}-\frac{8416146653}{4134156780950}a^{8}+\frac{35828749214}{2067078390475}a^{7}-\frac{954206142793}{4134156780950}a^{6}+\frac{537931247673}{826831356190}a^{5}-\frac{5462798981296}{2067078390475}a^{4}+\frac{16996548768467}{4134156780950}a^{3}-\frac{6944743047457}{590593825850}a^{2}+\frac{1130790633778}{59059382585}a-\frac{129161829397}{16874109310}$, $\frac{3048951049181}{157097957676100}a^{11}-\frac{439961429507}{78548978838050}a^{10}+\frac{81484228819163}{157097957676100}a^{9}+\frac{27322556130819}{157097957676100}a^{8}+\frac{40678199749539}{15709795767610}a^{7}-\frac{27\!\cdots\!31}{157097957676100}a^{6}+\frac{73\!\cdots\!67}{157097957676100}a^{5}-\frac{17\!\cdots\!07}{78548978838050}a^{4}+\frac{71\!\cdots\!07}{22442565382300}a^{3}-\frac{10\!\cdots\!87}{897702615292}a^{2}+\frac{398433981253291}{320608076890}a-\frac{57951046818723}{128243230756}$, $\frac{204599557381}{157097957676100}a^{11}+\frac{12302025497}{15709795767610}a^{10}+\frac{5878652704143}{157097957676100}a^{9}+\frac{1476658871703}{31419591535220}a^{8}+\frac{24240704568223}{78548978838050}a^{7}-\frac{134105549171547}{157097957676100}a^{6}+\frac{131088836977111}{31419591535220}a^{5}-\frac{994683239482813}{78548978838050}a^{4}+\frac{110440576824567}{4488513076460}a^{3}-\frac{15\!\cdots\!51}{22442565382300}a^{2}+\frac{24435142464733}{320608076890}a-\frac{15823012430471}{641216153780}$, $\frac{10\!\cdots\!81}{157097957676100}a^{11}+\frac{578061798402106}{39274489419025}a^{10}+\frac{27\!\cdots\!89}{157097957676100}a^{9}+\frac{23\!\cdots\!11}{157097957676100}a^{8}+\frac{37\!\cdots\!93}{39274489419025}a^{7}-\frac{83\!\cdots\!69}{157097957676100}a^{6}+\frac{20\!\cdots\!79}{157097957676100}a^{5}-\frac{26\!\cdots\!63}{39274489419025}a^{4}+\frac{11\!\cdots\!99}{157097957676100}a^{3}-\frac{76\!\cdots\!27}{22442565382300}a^{2}+\frac{27\!\cdots\!97}{1122128269115}a-\frac{22\!\cdots\!17}{641216153780}$
| 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: | \( 3969507.226057836 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 3969507.226057836 \cdot 4}{2\cdot\sqrt{7972425761140316000000}}\cr\approx \mathstrut & 3.48281722548910 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 27*x^10 - 10*x^9 + 129*x^8 - 987*x^7 + 3068*x^6 - 13047*x^5 + 24529*x^4 - 70220*x^3 + 105861*x^2 - 72135*x + 17885)
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 - x^11 + 27*x^10 - 10*x^9 + 129*x^8 - 987*x^7 + 3068*x^6 - 13047*x^5 + 24529*x^4 - 70220*x^3 + 105861*x^2 - 72135*x + 17885, 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 - x^11 + 27*x^10 - 10*x^9 + 129*x^8 - 987*x^7 + 3068*x^6 - 13047*x^5 + 24529*x^4 - 70220*x^3 + 105861*x^2 - 72135*x + 17885);
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 - x^11 + 27*x^10 - 10*x^9 + 129*x^8 - 987*x^7 + 3068*x^6 - 13047*x^5 + 24529*x^4 - 70220*x^3 + 105861*x^2 - 72135*x + 17885);
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
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 24 |
| The 9 conjugacy class representatives for $D_{12}$ |
| Character table for $D_{12}$ |
Intermediate fields
| \(\Q(\sqrt{85}) \), 3.1.140.1, 4.2.859775.1, 6.2.481474000.3 |
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
| Galois closure: | deg 24 |
| Degree 12 sibling: | deg 12 |
| 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 | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | R | R | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.12.0.1}{12} }$ | R | ${\href{/padicField/19.2.0.1}{2} }^{5}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(5\)
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(17\)
| 17.12.9.3 | $x^{12} + 289 x^{4} - 68782$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.35.2t1.a.a | $1$ | $ 5 \cdot 7 $ | \(\Q(\sqrt{-35}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.119.2t1.a.a | $1$ | $ 7 \cdot 17 $ | \(\Q(\sqrt{-119}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.85.2t1.a.a | $1$ | $ 5 \cdot 17 $ | \(\Q(\sqrt{85}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 2.140.3t2.a.a | $2$ | $ 2^{2} \cdot 5 \cdot 7 $ | 3.1.140.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.40460.6t3.c.a | $2$ | $ 2^{2} \cdot 5 \cdot 7 \cdot 17^{2}$ | 6.2.481474000.3 | $D_{6}$ (as 6T3) | $1$ | $0$ |
| * | 2.10115.4t3.f.a | $2$ | $ 5 \cdot 7 \cdot 17^{2}$ | 4.2.859775.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
| * | 2.40460.12t12.a.a | $2$ | $ 2^{2} \cdot 5 \cdot 7 \cdot 17^{2}$ | 12.2.7972425761140316000000.1 | $D_{12}$ (as 12T12) | $1$ | $0$ |
| * | 2.40460.12t12.a.b | $2$ | $ 2^{2} \cdot 5 \cdot 7 \cdot 17^{2}$ | 12.2.7972425761140316000000.1 | $D_{12}$ (as 12T12) | $1$ | $0$ |
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 *.