Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 732*x^10 + 200934*x^8 + 25421872*x^6 + 1453813305*x^4 + 30405466836*x^2 + 51520374361)
gp: K = bnfinit(y^12 + 732*y^10 + 200934*y^8 + 25421872*y^6 + 1453813305*y^4 + 30405466836*y^2 + 51520374361, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 732*x^10 + 200934*x^8 + 25421872*x^6 + 1453813305*x^4 + 30405466836*x^2 + 51520374361);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + 732*x^10 + 200934*x^8 + 25421872*x^6 + 1453813305*x^4 + 30405466836*x^2 + 51520374361)
\( x^{12} + 732x^{10} + 200934x^{8} + 25421872x^{6} + 1453813305x^{4} + 30405466836x^{2} + 51520374361 \)
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: |
\(334874047209198762552459264\)
\(\medspace = 2^{24}\cdot 3^{18}\cdot 61^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(162.33\) | 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: | $2^{2}3^{3/2}61^{1/2}\approx 162.3329911016242$ | ||
| Ramified primes: |
\(2\), \(3\), \(61\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(4392=2^{3}\cdot 3^{2}\cdot 61\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4392}(1,·)$, $\chi_{4392}(1219,·)$, $\chi_{4392}(1829,·)$, $\chi_{4392}(365,·)$, $\chi_{4392}(4271,·)$, $\chi_{4392}(2929,·)$, $\chi_{4392}(4147,·)$, $\chi_{4392}(2807,·)$, $\chi_{4392}(1465,·)$, $\chi_{4392}(2683,·)$, $\chi_{4392}(3293,·)$, $\chi_{4392}(1343,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-122}) \), \(\Q(\sqrt{-366}) \), 6.0.762481838592.7$^{3}$, 6.0.2287445515776.5$^{3}$, 12.0.334874047209198762552459264.5$^{24}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{61}a^{2}$, $\frac{1}{61}a^{3}$, $\frac{1}{3721}a^{4}$, $\frac{1}{3721}a^{5}$, $\frac{1}{226981}a^{6}$, $\frac{1}{226981}a^{7}$, $\frac{1}{13845841}a^{8}$, $\frac{1}{13845841}a^{9}$, $\frac{1}{844596301}a^{10}$, $\frac{1}{844596301}a^{11}$
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
$C_{42}\times C_{27090}$, which has order $1137780$ (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: |
$\frac{1}{13845841}a^{8}+\frac{8}{226981}a^{6}+\frac{19}{3721}a^{4}+\frac{12}{61}a^{2}$, $\frac{1}{3721}a^{4}+\frac{4}{61}a^{2}+2$, $\frac{1}{844596301}a^{10}+\frac{10}{13845841}a^{8}+\frac{35}{226981}a^{6}+\frac{50}{3721}a^{4}+\frac{25}{61}a^{2}+1$, $\frac{1}{61}a^{2}+3$, $\frac{1}{844596301}a^{10}+\frac{10}{13845841}a^{8}+\frac{35}{226981}a^{6}+\frac{50}{3721}a^{4}+\frac{24}{61}a^{2}-1$
| 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: | \( 325.67540279491664 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 325.67540279491664 \cdot 1137780}{2\cdot\sqrt{334874047209198762552459264}}\cr\approx \mathstrut & 0.622947896068877 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 + 732*x^10 + 200934*x^8 + 25421872*x^6 + 1453813305*x^4 + 30405466836*x^2 + 51520374361)
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 + 732*x^10 + 200934*x^8 + 25421872*x^6 + 1453813305*x^4 + 30405466836*x^2 + 51520374361, 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 + 732*x^10 + 200934*x^8 + 25421872*x^6 + 1453813305*x^4 + 30405466836*x^2 + 51520374361);
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 + 732*x^10 + 200934*x^8 + 25421872*x^6 + 1453813305*x^4 + 30405466836*x^2 + 51520374361);
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
$C_2\times C_6$ (as 12T2):
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)
| An abelian group of order 12 |
| The 12 conjugacy class representatives for $C_6\times C_2$ |
| Character table for $C_6\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-122}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-366}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{3}, \sqrt{-122})\), 6.0.762481838592.7, \(\Q(\zeta_{36})^+\), 6.0.2287445515776.5 |
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]
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.1.0.1}{1} }^{12}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.24.318 | $x^{12} + 10 x^{10} + 12 x^{9} + 110 x^{8} + 80 x^{7} + 752 x^{6} + 512 x^{5} + 1636 x^{4} + 1504 x^{3} + 1224 x^{2} + 1008 x - 648$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $[2, 3]^{3}$ |
|
\(3\)
| 3.6.9.3 | $x^{6} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
| 3.6.9.3 | $x^{6} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
|
\(61\)
| 61.6.3.2 | $x^{6} + 26047 x^{2} - 13391879$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 61.6.3.2 | $x^{6} + 26047 x^{2} - 13391879$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |