Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 16*x^10 - 30*x^9 + 46*x^8 - 42*x^7 + 66*x^6 - 30*x^5 + 81*x^4 + 46*x^3 + 98*x^2 + 60*x + 36)
gp: K = bnfinit(y^12 - 4*y^11 + 16*y^10 - 30*y^9 + 46*y^8 - 42*y^7 + 66*y^6 - 30*y^5 + 81*y^4 + 46*y^3 + 98*y^2 + 60*y + 36, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 16*x^10 - 30*x^9 + 46*x^8 - 42*x^7 + 66*x^6 - 30*x^5 + 81*x^4 + 46*x^3 + 98*x^2 + 60*x + 36);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 + 16*x^10 - 30*x^9 + 46*x^8 - 42*x^7 + 66*x^6 - 30*x^5 + 81*x^4 + 46*x^3 + 98*x^2 + 60*x + 36)
\( x^{12} - 4 x^{11} + 16 x^{10} - 30 x^{9} + 46 x^{8} - 42 x^{7} + 66 x^{6} - 30 x^{5} + 81 x^{4} + \cdots + 36 \)
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: |
\(3376439254122496\)
\(\medspace = 2^{16}\cdot 61^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.68\) | 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: | $2^{3/2}61^{1/2}\approx 22.090722034374522$ | ||
| Ramified primes: |
\(2\), \(61\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{6}a^{9}+\frac{1}{6}a^{8}+\frac{1}{6}a^{7}-\frac{1}{6}a^{5}+\frac{1}{6}a^{4}+\frac{1}{6}a^{3}-\frac{1}{3}a$, $\frac{1}{12}a^{10}-\frac{1}{4}a^{8}+\frac{1}{6}a^{7}-\frac{1}{12}a^{6}-\frac{1}{3}a^{5}-\frac{1}{4}a^{4}+\frac{1}{6}a^{3}-\frac{1}{6}a^{2}-\frac{1}{3}a$, $\frac{1}{50009592}a^{11}-\frac{1140521}{50009592}a^{10}-\frac{1135247}{16669864}a^{9}-\frac{6182029}{50009592}a^{8}+\frac{2654269}{50009592}a^{7}+\frac{50353}{50009592}a^{6}-\frac{4938421}{50009592}a^{5}-\frac{10535101}{50009592}a^{4}+\frac{125714}{2083733}a^{3}+\frac{3908599}{8334932}a^{2}-\frac{1435886}{6251199}a-\frac{864993}{4167466}$
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
$C_{4}$, which has order $4$
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: |
$\frac{172807}{50009592}a^{11}-\frac{2210375}{50009592}a^{10}+\frac{10333397}{50009592}a^{9}-\frac{34655759}{50009592}a^{8}+\frac{23878193}{16669864}a^{7}-\frac{100337321}{50009592}a^{6}+\frac{28883063}{16669864}a^{5}-\frac{81695591}{50009592}a^{4}+\frac{8195485}{6251199}a^{3}-\frac{12952023}{8334932}a^{2}-\frac{4393828}{6251199}a-\frac{2342329}{4167466}$, $\frac{334065}{16669864}a^{11}-\frac{1736281}{16669864}a^{10}+\frac{21401825}{50009592}a^{9}-\frac{51894619}{50009592}a^{8}+\frac{93256907}{50009592}a^{7}-\frac{40392491}{16669864}a^{6}+\frac{155146417}{50009592}a^{5}-\frac{142139131}{50009592}a^{4}+\frac{34251635}{12502398}a^{3}-\frac{10454565}{8334932}a^{2}+\frac{11161093}{6251199}a-\frac{387111}{4167466}$, $\frac{1559869}{25004796}a^{11}-\frac{1256635}{4167466}a^{10}+\frac{29987561}{25004796}a^{9}-\frac{33117733}{12502398}a^{8}+\frac{105325867}{25004796}a^{7}-\frac{55292897}{12502398}a^{6}+\frac{133547773}{25004796}a^{5}-\frac{55462627}{12502398}a^{4}+\frac{24458805}{4167466}a^{3}-\frac{1529794}{6251199}a^{2}+\frac{14834333}{6251199}a-\frac{303893}{2083733}$, $\frac{1253023}{50009592}a^{11}-\frac{1368887}{16669864}a^{10}+\frac{13378957}{50009592}a^{9}-\frac{3202719}{16669864}a^{8}-\frac{6491947}{16669864}a^{7}+\frac{85548665}{50009592}a^{6}-\frac{68240155}{50009592}a^{5}+\frac{29088073}{16669864}a^{4}+\frac{6005057}{12502398}a^{3}+\frac{66908167}{25004796}a^{2}+\frac{13577870}{6251199}a+\frac{5931043}{4167466}$, $\frac{1771753}{25004796}a^{11}-\frac{4756699}{25004796}a^{10}+\frac{17201815}{25004796}a^{9}-\frac{3645439}{8334932}a^{8}-\frac{5432153}{25004796}a^{7}+\frac{13936269}{8334932}a^{6}+\frac{10888057}{8334932}a^{5}+\frac{16665161}{8334932}a^{4}+\frac{5326423}{4167466}a^{3}+\frac{127116851}{12502398}a^{2}+\frac{55262540}{6251199}a+\frac{8757708}{2083733}$
| 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: | \( 618.75227265 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 618.75227265 \cdot 4}{2\cdot\sqrt{3376439254122496}}\cr\approx \mathstrut & 1.3103778475 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 16*x^10 - 30*x^9 + 46*x^8 - 42*x^7 + 66*x^6 - 30*x^5 + 81*x^4 + 46*x^3 + 98*x^2 + 60*x + 36)
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 + 16*x^10 - 30*x^9 + 46*x^8 - 42*x^7 + 66*x^6 - 30*x^5 + 81*x^4 + 46*x^3 + 98*x^2 + 60*x + 36, 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 + 16*x^10 - 30*x^9 + 46*x^8 - 42*x^7 + 66*x^6 - 30*x^5 + 81*x^4 + 46*x^3 + 98*x^2 + 60*x + 36);
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 + 16*x^10 - 30*x^9 + 46*x^8 - 42*x^7 + 66*x^6 - 30*x^5 + 81*x^4 + 46*x^3 + 98*x^2 + 60*x + 36);
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 5 conjugacy class representatives for $S_4$ |
| Character table for $S_4$ |
Intermediate fields
| \(\Q(\sqrt{-61}) \), 3.1.244.1 x3, 6.0.58107136.1, 6.2.238144.1, 6.0.14526784.1 |
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 4 sibling: | 4.2.976.1 |
| Degree 6 siblings: | 6.2.238144.1, 6.0.58107136.1 |
| Degree 8 sibling: | 8.0.56712564736.2 |
| Degree 12 sibling: | 12.2.55351463182336.2 |
| Minimal sibling: | 4.2.976.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.1.0.1}{1} }^{12}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.6.9 | $x^{4} + 2 x^{3} + 6$ | $4$ | $1$ | $6$ | $D_{4}$ | $[2, 2]^{2}$ |
| 2.4.6.9 | $x^{4} + 2 x^{3} + 6$ | $4$ | $1$ | $6$ | $D_{4}$ | $[2, 2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
|
\(61\)
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |