Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 367*x^10 + 788*x^9 + 53059*x^8 - 36136*x^7 - 3642675*x^6 - 265142*x^5 + 128044633*x^4 + 49855354*x^3 - 2174288521*x^2 - 388898690*x + 17877187700)
gp: K = bnfinit(y^12 - 4*y^11 - 367*y^10 + 788*y^9 + 53059*y^8 - 36136*y^7 - 3642675*y^6 - 265142*y^5 + 128044633*y^4 + 49855354*y^3 - 2174288521*y^2 - 388898690*y + 17877187700, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 367*x^10 + 788*x^9 + 53059*x^8 - 36136*x^7 - 3642675*x^6 - 265142*x^5 + 128044633*x^4 + 49855354*x^3 - 2174288521*x^2 - 388898690*x + 17877187700);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 - 367*x^10 + 788*x^9 + 53059*x^8 - 36136*x^7 - 3642675*x^6 - 265142*x^5 + 128044633*x^4 + 49855354*x^3 - 2174288521*x^2 - 388898690*x + 17877187700)
\( x^{12} - 4 x^{11} - 367 x^{10} + 788 x^{9} + 53059 x^{8} - 36136 x^{7} - 3642675 x^{6} + \cdots + 17877187700 \)
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: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(912188385171866131114042868895361\)
\(\medspace = 7^{10}\cdot 11^{6}\cdot 67^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(558.05\) | 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: | $7^{5/6}11^{1/2}67^{5/6}\approx 558.050769739001$ | ||
| Ramified primes: |
\(7\), \(11\), \(67\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{32}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{12}a^{8}-\frac{1}{12}a^{7}+\frac{1}{12}a^{6}-\frac{1}{4}a^{5}-\frac{1}{12}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{12}a^{9}-\frac{1}{6}a^{6}+\frac{1}{6}a^{5}+\frac{1}{6}a^{4}-\frac{1}{4}a^{3}-\frac{1}{6}a^{2}-\frac{1}{2}a-\frac{1}{3}$, $\frac{1}{1280880}a^{10}-\frac{7957}{1280880}a^{9}-\frac{10067}{426960}a^{8}+\frac{27259}{320220}a^{7}-\frac{137317}{640440}a^{6}-\frac{22649}{213480}a^{5}+\frac{214477}{1280880}a^{4}-\frac{26093}{1280880}a^{3}-\frac{288073}{1280880}a^{2}+\frac{278719}{640440}a+\frac{28115}{64044}$, $\frac{1}{18\!\cdots\!60}a^{11}-\frac{77\!\cdots\!91}{30\!\cdots\!36}a^{10}+\frac{39\!\cdots\!43}{18\!\cdots\!16}a^{9}+\frac{21\!\cdots\!79}{18\!\cdots\!60}a^{8}+\frac{51\!\cdots\!09}{92\!\cdots\!80}a^{7}-\frac{17\!\cdots\!17}{11\!\cdots\!85}a^{6}-\frac{34\!\cdots\!01}{18\!\cdots\!60}a^{5}-\frac{19\!\cdots\!24}{11\!\cdots\!85}a^{4}+\frac{66\!\cdots\!81}{30\!\cdots\!60}a^{3}-\frac{78\!\cdots\!43}{18\!\cdots\!60}a^{2}-\frac{14\!\cdots\!89}{30\!\cdots\!60}a-\frac{11\!\cdots\!09}{92\!\cdots\!08}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$, $5$ |
Class group and class number
$C_{3}\times C_{6}\times C_{6}\times C_{150612}$, which has order $16266096$ (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: | $5$ | 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: |
$\frac{13\!\cdots\!73}{10\!\cdots\!12}a^{11}+\frac{80\!\cdots\!39}{12\!\cdots\!44}a^{10}+\frac{49\!\cdots\!33}{12\!\cdots\!44}a^{9}-\frac{49\!\cdots\!13}{41\!\cdots\!48}a^{8}-\frac{14\!\cdots\!83}{30\!\cdots\!36}a^{7}+\frac{35\!\cdots\!93}{61\!\cdots\!72}a^{6}+\frac{12\!\cdots\!25}{68\!\cdots\!08}a^{5}-\frac{16\!\cdots\!85}{12\!\cdots\!44}a^{4}-\frac{43\!\cdots\!95}{12\!\cdots\!44}a^{3}+\frac{81\!\cdots\!13}{12\!\cdots\!44}a^{2}-\frac{71\!\cdots\!35}{61\!\cdots\!72}a-\frac{94\!\cdots\!47}{30\!\cdots\!36}$, $\frac{36\!\cdots\!69}{51\!\cdots\!60}a^{11}+\frac{62\!\cdots\!93}{30\!\cdots\!60}a^{10}-\frac{69\!\cdots\!51}{30\!\cdots\!60}a^{9}-\frac{79\!\cdots\!19}{10\!\cdots\!20}a^{8}+\frac{49\!\cdots\!31}{30\!\cdots\!36}a^{7}+\frac{16\!\cdots\!43}{15\!\cdots\!18}a^{6}+\frac{20\!\cdots\!93}{17\!\cdots\!20}a^{5}-\frac{34\!\cdots\!13}{61\!\cdots\!72}a^{4}-\frac{94\!\cdots\!81}{61\!\cdots\!72}a^{3}+\frac{32\!\cdots\!49}{30\!\cdots\!60}a^{2}+\frac{72\!\cdots\!19}{15\!\cdots\!80}a+\frac{38\!\cdots\!21}{15\!\cdots\!18}$, $\frac{2478590140893}{14\!\cdots\!28}a^{11}-\frac{142779383515401}{56\!\cdots\!12}a^{10}-\frac{28\!\cdots\!11}{56\!\cdots\!12}a^{9}+\frac{41\!\cdots\!73}{56\!\cdots\!12}a^{8}+\frac{85\!\cdots\!05}{14\!\cdots\!28}a^{7}-\frac{23\!\cdots\!75}{28\!\cdots\!56}a^{6}-\frac{98\!\cdots\!07}{28\!\cdots\!56}a^{5}+\frac{23\!\cdots\!67}{56\!\cdots\!12}a^{4}+\frac{54\!\cdots\!33}{56\!\cdots\!12}a^{3}-\frac{47\!\cdots\!79}{56\!\cdots\!12}a^{2}-\frac{26\!\cdots\!55}{28\!\cdots\!56}a+\frac{12\!\cdots\!81}{14\!\cdots\!28}$, $\frac{72\!\cdots\!71}{25\!\cdots\!30}a^{11}-\frac{16\!\cdots\!19}{30\!\cdots\!60}a^{10}+\frac{35\!\cdots\!83}{30\!\cdots\!60}a^{9}+\frac{22\!\cdots\!77}{10\!\cdots\!20}a^{8}-\frac{23\!\cdots\!33}{15\!\cdots\!18}a^{7}-\frac{93\!\cdots\!37}{30\!\cdots\!36}a^{6}+\frac{11\!\cdots\!71}{17\!\cdots\!20}a^{5}+\frac{10\!\cdots\!85}{61\!\cdots\!72}a^{4}-\frac{19\!\cdots\!93}{61\!\cdots\!72}a^{3}-\frac{10\!\cdots\!97}{30\!\cdots\!60}a^{2}+\frac{10\!\cdots\!63}{15\!\cdots\!80}a+\frac{53\!\cdots\!45}{15\!\cdots\!18}$, $\frac{18\!\cdots\!11}{42\!\cdots\!55}a^{11}+\frac{31\!\cdots\!51}{20\!\cdots\!40}a^{10}-\frac{33\!\cdots\!07}{20\!\cdots\!40}a^{9}-\frac{55\!\cdots\!33}{68\!\cdots\!80}a^{8}+\frac{56\!\cdots\!44}{25\!\cdots\!53}a^{7}+\frac{27\!\cdots\!19}{20\!\cdots\!24}a^{6}-\frac{36\!\cdots\!17}{34\!\cdots\!40}a^{5}-\frac{32\!\cdots\!17}{41\!\cdots\!48}a^{4}+\frac{69\!\cdots\!49}{41\!\cdots\!48}a^{3}+\frac{31\!\cdots\!53}{20\!\cdots\!40}a^{2}-\frac{17\!\cdots\!87}{10\!\cdots\!20}a-\frac{15\!\cdots\!59}{10\!\cdots\!12}$
| 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: | \( 1991588.7507786483 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1991588.7507786483 \cdot 16266096}{2\cdot\sqrt{912188385171866131114042868895361}}\cr\approx \mathstrut & 32.9981764960024 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 367*x^10 + 788*x^9 + 53059*x^8 - 36136*x^7 - 3642675*x^6 - 265142*x^5 + 128044633*x^4 + 49855354*x^3 - 2174288521*x^2 - 388898690*x + 17877187700)
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 - 4*x^11 - 367*x^10 + 788*x^9 + 53059*x^8 - 36136*x^7 - 3642675*x^6 - 265142*x^5 + 128044633*x^4 + 49855354*x^3 - 2174288521*x^2 - 388898690*x + 17877187700, 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 - 4*x^11 - 367*x^10 + 788*x^9 + 53059*x^8 - 36136*x^7 - 3642675*x^6 - 265142*x^5 + 128044633*x^4 + 49855354*x^3 - 2174288521*x^2 - 388898690*x + 17877187700);
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 - 4*x^11 - 367*x^10 + 788*x^9 + 53059*x^8 - 36136*x^7 - 3642675*x^6 - 265142*x^5 + 128044633*x^4 + 49855354*x^3 - 2174288521*x^2 - 388898690*x + 17877187700);
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
$C_6\times S_3$ (as 12T18):
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 36 |
| The 18 conjugacy class representatives for $C_6\times S_3$ |
| Character table for $C_6\times S_3$ |
Intermediate fields
| \(\Q(\sqrt{-5159}) \), \(\Q(\sqrt{469}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{469})\), 6.6.22691552673349.3 |
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: | data not computed |
| Degree 18 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 | ${\href{/padicField/2.2.0.1}{2} }^{6}$ | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{6}$ | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{6}$ | R | R | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.12.10.3 | $x^{12} - 1176$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |
|
\(11\)
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(67\)
| 67.6.5.6 | $x^{6} + 402$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 67.6.5.6 | $x^{6} + 402$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |