Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 264*x^10 - 1332*x^9 + 41472*x^8 + 346320*x^7 - 451748*x^6 - 3516480*x^5 + 219344376*x^4 + 1374245712*x^3 + 4479083712*x^2 + 36662777856*x + 412946670736)
gp: K = bnfinit(y^12 - 264*y^10 - 1332*y^9 + 41472*y^8 + 346320*y^7 - 451748*y^6 - 3516480*y^5 + 219344376*y^4 + 1374245712*y^3 + 4479083712*y^2 + 36662777856*y + 412946670736, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 264*x^10 - 1332*x^9 + 41472*x^8 + 346320*x^7 - 451748*x^6 - 3516480*x^5 + 219344376*x^4 + 1374245712*x^3 + 4479083712*x^2 + 36662777856*x + 412946670736);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 264*x^10 - 1332*x^9 + 41472*x^8 + 346320*x^7 - 451748*x^6 - 3516480*x^5 + 219344376*x^4 + 1374245712*x^3 + 4479083712*x^2 + 36662777856*x + 412946670736)
\( x^{12} - 264 x^{10} - 1332 x^{9} + 41472 x^{8} + 346320 x^{7} - 451748 x^{6} - 3516480 x^{5} + \cdots + 412946670736 \)
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: |
\(122089905124794772482173239296000000\)
\(\medspace = 2^{22}\cdot 3^{18}\cdot 5^{6}\cdot 37^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(839.25\) | 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: | $2^{11/6}3^{3/2}5^{1/2}37^{5/6}\approx 839.2473521053773$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(37\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{12}a^{6}-\frac{1}{3}$, $\frac{1}{12}a^{7}-\frac{1}{3}a$, $\frac{1}{12}a^{8}-\frac{1}{3}a^{2}$, $\frac{1}{24}a^{9}-\frac{1}{6}a^{3}$, $\frac{1}{125596728}a^{10}-\frac{128609}{6977596}a^{9}+\frac{606137}{15699591}a^{8}+\frac{707317}{20932788}a^{7}-\frac{778315}{62798364}a^{6}+\frac{229632}{1744399}a^{5}+\frac{3344873}{31399182}a^{4}+\frac{199163}{3488798}a^{3}+\frac{971245}{15699591}a^{2}+\frac{2607539}{5233197}a-\frac{4142324}{15699591}$, $\frac{1}{18\!\cdots\!44}a^{11}+\frac{21\!\cdots\!47}{18\!\cdots\!44}a^{10}+\frac{21\!\cdots\!45}{22\!\cdots\!43}a^{9}+\frac{42\!\cdots\!97}{91\!\cdots\!72}a^{8}+\frac{62\!\cdots\!31}{22\!\cdots\!43}a^{7}-\frac{27\!\cdots\!89}{91\!\cdots\!72}a^{6}+\frac{56\!\cdots\!31}{45\!\cdots\!86}a^{5}+\frac{50\!\cdots\!09}{45\!\cdots\!86}a^{4}-\frac{13\!\cdots\!77}{45\!\cdots\!86}a^{3}-\frac{10\!\cdots\!11}{22\!\cdots\!43}a^{2}+\frac{49\!\cdots\!40}{22\!\cdots\!43}a+\frac{10\!\cdots\!11}{22\!\cdots\!43}$
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
$C_{6}\times C_{6}\times C_{1189032}$, which has order $42805152$ (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: |
$\frac{506759090925825}{84\!\cdots\!77}a^{11}-\frac{51\!\cdots\!58}{84\!\cdots\!77}a^{10}-\frac{20\!\cdots\!15}{16\!\cdots\!54}a^{9}+\frac{13\!\cdots\!81}{16\!\cdots\!54}a^{8}+\frac{16\!\cdots\!24}{84\!\cdots\!77}a^{7}-\frac{56\!\cdots\!11}{16\!\cdots\!54}a^{6}-\frac{43\!\cdots\!40}{84\!\cdots\!77}a^{5}+\frac{61\!\cdots\!67}{84\!\cdots\!77}a^{4}+\frac{48\!\cdots\!18}{84\!\cdots\!77}a^{3}-\frac{14\!\cdots\!56}{84\!\cdots\!77}a^{2}+\frac{13\!\cdots\!88}{84\!\cdots\!77}a+\frac{11\!\cdots\!17}{84\!\cdots\!77}$, $\frac{79\!\cdots\!11}{15\!\cdots\!62}a^{11}-\frac{13\!\cdots\!07}{76\!\cdots\!81}a^{10}+\frac{32\!\cdots\!29}{30\!\cdots\!24}a^{9}+\frac{12\!\cdots\!83}{10\!\cdots\!08}a^{8}-\frac{55\!\cdots\!98}{76\!\cdots\!81}a^{7}-\frac{20\!\cdots\!61}{10\!\cdots\!08}a^{6}-\frac{16\!\cdots\!64}{76\!\cdots\!81}a^{5}-\frac{22\!\cdots\!79}{15\!\cdots\!62}a^{4}-\frac{73\!\cdots\!83}{76\!\cdots\!81}a^{3}-\frac{11\!\cdots\!94}{25\!\cdots\!27}a^{2}-\frac{14\!\cdots\!68}{76\!\cdots\!81}a+\frac{19\!\cdots\!80}{25\!\cdots\!27}$, $\frac{96\!\cdots\!01}{15\!\cdots\!62}a^{11}-\frac{31\!\cdots\!27}{30\!\cdots\!24}a^{10}-\frac{24\!\cdots\!79}{30\!\cdots\!24}a^{9}+\frac{49\!\cdots\!55}{30\!\cdots\!24}a^{8}+\frac{11\!\cdots\!16}{76\!\cdots\!81}a^{7}-\frac{65\!\cdots\!11}{30\!\cdots\!24}a^{6}+\frac{26\!\cdots\!86}{76\!\cdots\!81}a^{5}+\frac{22\!\cdots\!17}{15\!\cdots\!62}a^{4}-\frac{40\!\cdots\!71}{76\!\cdots\!81}a^{3}-\frac{54\!\cdots\!50}{76\!\cdots\!81}a^{2}+\frac{52\!\cdots\!64}{76\!\cdots\!81}a-\frac{42\!\cdots\!16}{76\!\cdots\!81}$, $\frac{13\!\cdots\!45}{10\!\cdots\!08}a^{11}-\frac{22\!\cdots\!11}{20\!\cdots\!16}a^{10}-\frac{76\!\cdots\!62}{25\!\cdots\!27}a^{9}+\frac{20\!\cdots\!15}{15\!\cdots\!62}a^{8}+\frac{12\!\cdots\!00}{25\!\cdots\!27}a^{7}+\frac{18\!\cdots\!79}{76\!\cdots\!81}a^{6}-\frac{48\!\cdots\!97}{25\!\cdots\!27}a^{5}+\frac{45\!\cdots\!91}{50\!\cdots\!54}a^{4}+\frac{44\!\cdots\!06}{25\!\cdots\!27}a^{3}+\frac{30\!\cdots\!33}{76\!\cdots\!81}a^{2}-\frac{43\!\cdots\!66}{25\!\cdots\!27}a+\frac{18\!\cdots\!51}{76\!\cdots\!81}$, $\frac{10\!\cdots\!09}{15\!\cdots\!62}a^{11}-\frac{39\!\cdots\!87}{61\!\cdots\!48}a^{10}-\frac{45\!\cdots\!53}{30\!\cdots\!24}a^{9}+\frac{19\!\cdots\!96}{25\!\cdots\!27}a^{8}+\frac{18\!\cdots\!83}{76\!\cdots\!81}a^{7}-\frac{13\!\cdots\!51}{10\!\cdots\!08}a^{6}-\frac{49\!\cdots\!83}{76\!\cdots\!81}a^{5}+\frac{10\!\cdots\!31}{15\!\cdots\!62}a^{4}+\frac{60\!\cdots\!43}{76\!\cdots\!81}a^{3}+\frac{19\!\cdots\!56}{25\!\cdots\!27}a^{2}+\frac{90\!\cdots\!30}{76\!\cdots\!81}a+\frac{54\!\cdots\!24}{25\!\cdots\!27}$
| 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: | \( 207114.39572456083 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 207114.39572456083 \cdot 42805152}{2\cdot\sqrt{122089905124794772482173239296000000}}\cr\approx \mathstrut & 0.780576848158092 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 264*x^10 - 1332*x^9 + 41472*x^8 + 346320*x^7 - 451748*x^6 - 3516480*x^5 + 219344376*x^4 + 1374245712*x^3 + 4479083712*x^2 + 36662777856*x + 412946670736)
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 - 264*x^10 - 1332*x^9 + 41472*x^8 + 346320*x^7 - 451748*x^6 - 3516480*x^5 + 219344376*x^4 + 1374245712*x^3 + 4479083712*x^2 + 36662777856*x + 412946670736, 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 - 264*x^10 - 1332*x^9 + 41472*x^8 + 346320*x^7 - 451748*x^6 - 3516480*x^5 + 219344376*x^4 + 1374245712*x^3 + 4479083712*x^2 + 36662777856*x + 412946670736);
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 - 264*x^10 - 1332*x^9 + 41472*x^8 + 346320*x^7 - 451748*x^6 - 3516480*x^5 + 219344376*x^4 + 1374245712*x^3 + 4479083712*x^2 + 36662777856*x + 412946670736);
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
| \(\Q(\sqrt{37}) \), \(\Q(\sqrt{-1110}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-30}, \sqrt{37})\), 6.6.7279451230032.1 |
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
| 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 | R | R | R | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.22.27 | $x^{12} + 8 x^{9} + 16 x^{8} + 4 x^{7} + 2 x^{6} + 40 x^{5} + 104 x^{4} - 48 x^{3} - 12 x^{2} + 56 x + 196$ | $6$ | $2$ | $22$ | $C_6\times S_3$ | $[3]_{3}^{6}$ |
|
\(3\)
| 3.6.9.9 | $x^{6} + 6 x^{4} + 21$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
| 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
|
\(5\)
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(37\)
| 37.6.5.1 | $x^{6} + 37$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 37.6.5.1 | $x^{6} + 37$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |