Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 27*x^10 - 20*x^9 + 324*x^8 - 210*x^7 + 1968*x^6 - 630*x^5 + 5661*x^4 - 520*x^3 + 7380*x^2 + 3280)
gp: K = bnfinit(y^12 + 27*y^10 - 20*y^9 + 324*y^8 - 210*y^7 + 1968*y^6 - 630*y^5 + 5661*y^4 - 520*y^3 + 7380*y^2 + 3280, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 27*x^10 - 20*x^9 + 324*x^8 - 210*x^7 + 1968*x^6 - 630*x^5 + 5661*x^4 - 520*x^3 + 7380*x^2 + 3280);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + 27*x^10 - 20*x^9 + 324*x^8 - 210*x^7 + 1968*x^6 - 630*x^5 + 5661*x^4 - 520*x^3 + 7380*x^2 + 3280)
\( x^{12} + 27 x^{10} - 20 x^{9} + 324 x^{8} - 210 x^{7} + 1968 x^{6} - 630 x^{5} + 5661 x^{4} + \cdots + 3280 \)
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: |
\(446676160500000000\)
\(\medspace = 2^{8}\cdot 3^{12}\cdot 5^{9}\cdot 41^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.57\) | 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: | $2\cdot 3^{79/54}5^{3/4}41^{2/3}\approx 396.69567102648426$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(41\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{9}+\frac{3}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{43\!\cdots\!64}a^{11}-\frac{999690133719313}{21\!\cdots\!32}a^{10}-\frac{25\!\cdots\!89}{43\!\cdots\!64}a^{9}+\frac{69\!\cdots\!43}{21\!\cdots\!32}a^{8}+\frac{52\!\cdots\!17}{10\!\cdots\!16}a^{7}-\frac{10\!\cdots\!73}{21\!\cdots\!32}a^{6}-\frac{54\!\cdots\!39}{10\!\cdots\!16}a^{5}-\frac{38\!\cdots\!87}{21\!\cdots\!32}a^{4}-\frac{64\!\cdots\!31}{43\!\cdots\!64}a^{3}-\frac{33\!\cdots\!73}{21\!\cdots\!32}a^{2}+\frac{972370001877287}{10\!\cdots\!16}a+\frac{951882889208629}{27\!\cdots\!79}$
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 $1$
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: |
\( \frac{15500786844339}{21949826369785432} a^{11} + \frac{27751140888123}{21949826369785432} a^{10} + \frac{375609142391647}{21949826369785432} a^{9} + \frac{406838397290229}{21949826369785432} a^{8} + \frac{1713425502939873}{10974913184892716} a^{7} + \frac{2848304050558959}{10974913184892716} a^{6} + \frac{7185128607484389}{10974913184892716} a^{5} + \frac{20488579510413213}{10974913184892716} a^{4} + \frac{25054598797503653}{21949826369785432} a^{3} + \frac{98360570878469451}{21949826369785432} a^{2} + \frac{15072739802154837}{10974913184892716} a + \frac{19199617478949547}{5487456592446358} \)
(order $10$)
| 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: |
$\frac{61873534468611}{43\!\cdots\!64}a^{11}-\frac{24911570421327}{10\!\cdots\!16}a^{10}+\frac{15\!\cdots\!09}{43\!\cdots\!64}a^{9}-\frac{462227327967303}{54\!\cdots\!58}a^{8}+\frac{25\!\cdots\!39}{54\!\cdots\!58}a^{7}-\frac{18\!\cdots\!27}{21\!\cdots\!32}a^{6}+\frac{14\!\cdots\!31}{54\!\cdots\!58}a^{5}-\frac{79\!\cdots\!81}{21\!\cdots\!32}a^{4}+\frac{25\!\cdots\!15}{43\!\cdots\!64}a^{3}-\frac{61\!\cdots\!75}{10\!\cdots\!16}a^{2}+\frac{98\!\cdots\!11}{27\!\cdots\!79}a-\frac{13\!\cdots\!01}{54\!\cdots\!58}$, $\frac{241557734742275}{43\!\cdots\!64}a^{11}+\frac{264164236695787}{21\!\cdots\!32}a^{10}+\frac{62\!\cdots\!97}{43\!\cdots\!64}a^{9}+\frac{45\!\cdots\!79}{21\!\cdots\!32}a^{8}+\frac{15\!\cdots\!29}{10\!\cdots\!16}a^{7}+\frac{57\!\cdots\!37}{21\!\cdots\!32}a^{6}+\frac{74\!\cdots\!73}{10\!\cdots\!16}a^{5}+\frac{40\!\cdots\!35}{21\!\cdots\!32}a^{4}+\frac{69\!\cdots\!47}{43\!\cdots\!64}a^{3}+\frac{11\!\cdots\!83}{21\!\cdots\!32}a^{2}+\frac{17\!\cdots\!59}{10\!\cdots\!16}a+\frac{12\!\cdots\!28}{27\!\cdots\!79}$, $\frac{17884785338191}{54\!\cdots\!58}a^{11}+\frac{306516086871749}{21\!\cdots\!32}a^{10}+\frac{693721799736925}{10\!\cdots\!16}a^{9}+\frac{50\!\cdots\!95}{21\!\cdots\!32}a^{8}+\frac{32\!\cdots\!11}{10\!\cdots\!16}a^{7}+\frac{14\!\cdots\!53}{54\!\cdots\!58}a^{6}+\frac{92\!\cdots\!59}{10\!\cdots\!16}a^{5}+\frac{54\!\cdots\!05}{54\!\cdots\!58}a^{4}+\frac{21\!\cdots\!67}{10\!\cdots\!16}a^{3}+\frac{33\!\cdots\!81}{21\!\cdots\!32}a^{2}+\frac{16\!\cdots\!99}{10\!\cdots\!16}a+\frac{39\!\cdots\!43}{54\!\cdots\!58}$, $\frac{7109739667085}{54\!\cdots\!58}a^{11}-\frac{24588395389913}{27\!\cdots\!79}a^{10}+\frac{220645857332821}{54\!\cdots\!58}a^{9}-\frac{618944648395724}{27\!\cdots\!79}a^{8}+\frac{18\!\cdots\!17}{27\!\cdots\!79}a^{7}-\frac{64\!\cdots\!97}{27\!\cdots\!79}a^{6}+\frac{11\!\cdots\!35}{27\!\cdots\!79}a^{5}-\frac{26\!\cdots\!86}{27\!\cdots\!79}a^{4}+\frac{54\!\cdots\!51}{54\!\cdots\!58}a^{3}-\frac{35\!\cdots\!67}{27\!\cdots\!79}a^{2}+\frac{16\!\cdots\!26}{27\!\cdots\!79}a-\frac{37\!\cdots\!71}{27\!\cdots\!79}$, $\frac{14626665921023}{54\!\cdots\!58}a^{11}-\frac{91156048538759}{21\!\cdots\!32}a^{10}+\frac{722833290373991}{10\!\cdots\!16}a^{9}-\frac{33\!\cdots\!65}{21\!\cdots\!32}a^{8}+\frac{86\!\cdots\!33}{10\!\cdots\!16}a^{7}-\frac{81\!\cdots\!55}{54\!\cdots\!58}a^{6}+\frac{45\!\cdots\!95}{10\!\cdots\!16}a^{5}-\frac{30\!\cdots\!07}{54\!\cdots\!58}a^{4}+\frac{77\!\cdots\!55}{10\!\cdots\!16}a^{3}-\frac{17\!\cdots\!75}{21\!\cdots\!32}a^{2}+\frac{35\!\cdots\!45}{10\!\cdots\!16}a-\frac{20\!\cdots\!39}{54\!\cdots\!58}$
| 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: | \( 27699.7184504 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 27699.7184504 \cdot 1}{10\cdot\sqrt{446676160500000000}}\cr\approx \mathstrut & 0.255010567485 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 + 27*x^10 - 20*x^9 + 324*x^8 - 210*x^7 + 1968*x^6 - 630*x^5 + 5661*x^4 - 520*x^3 + 7380*x^2 + 3280)
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 + 27*x^10 - 20*x^9 + 324*x^8 - 210*x^7 + 1968*x^6 - 630*x^5 + 5661*x^4 - 520*x^3 + 7380*x^2 + 3280, 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 + 27*x^10 - 20*x^9 + 324*x^8 - 210*x^7 + 1968*x^6 - 630*x^5 + 5661*x^4 - 520*x^3 + 7380*x^2 + 3280);
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 + 27*x^10 - 20*x^9 + 324*x^8 - 210*x^7 + 1968*x^6 - 630*x^5 + 5661*x^4 - 520*x^3 + 7380*x^2 + 3280);
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
$C_3^3:(C_4\times S_3)$ (as 12T170):
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 648 |
| The 30 conjugacy class representatives for $C_3^3:(C_4\times S_3)$ |
| Character table for $C_3^3:(C_4\times S_3)$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\) |
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
| Degree 12 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 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 | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }$ | R | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
|
\(3\)
| 3.12.12.7 | $x^{12} + 6 x^{11} + 24 x^{10} + 48 x^{9} + 576 x^{8} + 5508 x^{7} + 19710 x^{6} + 29538 x^{5} + 13608 x^{4} - 3456 x^{3} + 1458 x^{2} - 324 x + 81$ | $3$ | $4$ | $12$ | 12T119 | $[3/2, 3/2, 3/2]_{2}^{4}$ |
|
\(5\)
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
|
\(41\)
| $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.3.2.1 | $x^{3} + 41$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |