Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 3*x^10 - 7*x^9 + 18*x^8 - 18*x^7 - 31*x^6 + 12*x^5 + 72*x^4 + 60*x^3 - 108*x^2 - 27*x + 9)
gp: K = bnfinit(y^12 - 3*y^11 + 3*y^10 - 7*y^9 + 18*y^8 - 18*y^7 - 31*y^6 + 12*y^5 + 72*y^4 + 60*y^3 - 108*y^2 - 27*y + 9, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 + 3*x^10 - 7*x^9 + 18*x^8 - 18*x^7 - 31*x^6 + 12*x^5 + 72*x^4 + 60*x^3 - 108*x^2 - 27*x + 9);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 3*x^11 + 3*x^10 - 7*x^9 + 18*x^8 - 18*x^7 - 31*x^6 + 12*x^5 + 72*x^4 + 60*x^3 - 108*x^2 - 27*x + 9)
\( x^{12} - 3 x^{11} + 3 x^{10} - 7 x^{9} + 18 x^{8} - 18 x^{7} - 31 x^{6} + 12 x^{5} + 72 x^{4} + 60 x^{3} + \cdots + 9 \)
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: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1039043198361249\)
\(\medspace = 3^{16}\cdot 17^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.84\) | 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: | $3^{4/3}17^{1/2}\approx 17.839641950637386$ | ||
| Ramified primes: |
\(3\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{12}a^{9}-\frac{1}{4}a^{8}-\frac{1}{12}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{12}a^{3}+\frac{1}{4}$, $\frac{1}{12}a^{10}-\frac{1}{4}a^{8}-\frac{1}{12}a^{7}-\frac{1}{2}a^{5}+\frac{1}{6}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{47965404}a^{11}-\frac{1326425}{47965404}a^{10}+\frac{50469}{7994234}a^{9}+\frac{3141581}{47965404}a^{8}-\frac{7723909}{47965404}a^{7}-\frac{1215561}{15988468}a^{6}+\frac{17357219}{47965404}a^{5}-\frac{2731585}{11991351}a^{4}-\frac{1486483}{7994234}a^{3}+\frac{7763035}{15988468}a^{2}-\frac{3682383}{7994234}a-\frac{604740}{3997117}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $7$ | 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: |
$\frac{1929643}{47965404}a^{11}-\frac{4822261}{47965404}a^{10}+\frac{4366757}{47965404}a^{9}-\frac{6886505}{23982702}a^{8}+\frac{30958177}{47965404}a^{7}-\frac{6933308}{11991351}a^{6}-\frac{29076827}{23982702}a^{5}-\frac{26684477}{47965404}a^{4}+\frac{109564837}{47965404}a^{3}+\frac{50650285}{15988468}a^{2}-\frac{17682603}{7994234}a-\frac{11915073}{15988468}$, $\frac{308204}{11991351}a^{11}-\frac{5406749}{47965404}a^{10}+\frac{3599563}{23982702}a^{9}-\frac{11197937}{47965404}a^{8}+\frac{33332261}{47965404}a^{7}-\frac{21861427}{23982702}a^{6}-\frac{5192093}{11991351}a^{5}+\frac{17387789}{11991351}a^{4}+\frac{137122501}{47965404}a^{3}+\frac{148163}{3997117}a^{2}-\frac{82000417}{15988468}a-\frac{15502521}{15988468}$, $\frac{76823}{15988468}a^{11}-\frac{335165}{47965404}a^{10}-\frac{70509}{7994234}a^{9}-\frac{247297}{15988468}a^{8}+\frac{2466863}{47965404}a^{7}+\frac{808187}{15988468}a^{6}-\frac{4595963}{15988468}a^{5}-\frac{3732982}{11991351}a^{4}+\frac{4647543}{7994234}a^{3}+\frac{13355747}{15988468}a^{2}+\frac{748047}{7994234}a-\frac{6344621}{3997117}$, $\frac{228793}{15988468}a^{11}-\frac{732217}{11991351}a^{10}+\frac{770185}{11991351}a^{9}-\frac{914279}{7994234}a^{8}+\frac{4804405}{11991351}a^{7}-\frac{19959691}{47965404}a^{6}-\frac{5475173}{15988468}a^{5}+\frac{9102007}{11991351}a^{4}+\frac{100207763}{47965404}a^{3}+\frac{19389181}{15988468}a^{2}-\frac{62711147}{15988468}a-\frac{28866199}{15988468}$, $\frac{1885243}{47965404}a^{11}-\frac{5074139}{47965404}a^{10}+\frac{1908915}{15988468}a^{9}-\frac{8287289}{23982702}a^{8}+\frac{35867645}{47965404}a^{7}-\frac{6369617}{7994234}a^{6}-\frac{8637121}{11991351}a^{5}-\frac{27278767}{47965404}a^{4}+\frac{30090715}{15988468}a^{3}+\frac{43295195}{15988468}a^{2}-\frac{10974144}{3997117}a+\frac{3363605}{15988468}$, $\frac{141787}{15988468}a^{11}-\frac{411407}{47965404}a^{10}-\frac{1829471}{47965404}a^{9}+\frac{155971}{3997117}a^{8}-\frac{3349531}{47965404}a^{7}+\frac{8228989}{23982702}a^{6}-\frac{4228033}{3997117}a^{5}+\frac{12880127}{47965404}a^{4}+\frac{20294681}{47965404}a^{3}+\frac{26011531}{15988468}a^{2}-\frac{7864941}{7994234}a-\frac{5377771}{15988468}$, $\frac{799577}{15988468}a^{11}-\frac{913714}{11991351}a^{10}+\frac{1236443}{47965404}a^{9}-\frac{4302009}{15988468}a^{8}+\frac{10320359}{23982702}a^{7}-\frac{1200020}{11991351}a^{6}-\frac{8196140}{3997117}a^{5}-\frac{90423743}{47965404}a^{4}+\frac{17990777}{23982702}a^{3}+\frac{71734919}{15988468}a^{2}+\frac{1467007}{15988468}a-\frac{2863119}{3997117}$
| 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: | \( 989.16216967 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 989.16216967 \cdot 1}{2\cdot\sqrt{1039043198361249}}\cr\approx \mathstrut & 0.38261338333 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 3*x^10 - 7*x^9 + 18*x^8 - 18*x^7 - 31*x^6 + 12*x^5 + 72*x^4 + 60*x^3 - 108*x^2 - 27*x + 9)
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 - 3*x^11 + 3*x^10 - 7*x^9 + 18*x^8 - 18*x^7 - 31*x^6 + 12*x^5 + 72*x^4 + 60*x^3 - 108*x^2 - 27*x + 9, 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 - 3*x^11 + 3*x^10 - 7*x^9 + 18*x^8 - 18*x^7 - 31*x^6 + 12*x^5 + 72*x^4 + 60*x^3 - 108*x^2 - 27*x + 9);
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 - 3*x^11 + 3*x^10 - 7*x^9 + 18*x^8 - 18*x^7 - 31*x^6 + 12*x^5 + 72*x^4 + 60*x^3 - 108*x^2 - 27*x + 9);
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
$C_2\times A_4$ (as 12T7):
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 8 conjugacy class representatives for $A_4 \times C_2$ |
| Character table for $A_4 \times C_2$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), \(\Q(\zeta_{9})^+\), 6.6.32234193.1, 6.2.111537.1, 6.2.1896129.1 |
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: | deg 24 |
| Degree 6 sibling: | 6.2.111537.1 |
| Degree 8 sibling: | 8.0.547981281.1 |
| Degree 12 sibling: | 12.0.3595305184641.1 |
| Minimal sibling: | 6.2.111537.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.6.8.3 | $x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ |
| 3.6.8.3 | $x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ | |
|
\(17\)
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |