Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 189*x^10 + 15972*x^8 + 773671*x^6 + 22405545*x^4 + 361326234*x^2 + 2491307569)
gp: K = bnfinit(y^12 + 189*y^10 + 15972*y^8 + 773671*y^6 + 22405545*y^4 + 361326234*y^2 + 2491307569, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 189*x^10 + 15972*x^8 + 773671*x^6 + 22405545*x^4 + 361326234*x^2 + 2491307569);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + 189*x^10 + 15972*x^8 + 773671*x^6 + 22405545*x^4 + 361326234*x^2 + 2491307569)
\( x^{12} + 189x^{10} + 15972x^{8} + 773671x^{6} + 22405545x^{4} + 361326234x^{2} + 2491307569 \)
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: |
\(10492184911088211262672896\)
\(\medspace = 2^{12}\cdot 3^{18}\cdot 137^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(121.64\) | 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: | $2\cdot 3^{3/2}137^{1/2}\approx 121.63880959627976$ | ||
| Ramified primes: |
\(2\), \(3\), \(137\)
| 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: | \(4932=2^{2}\cdot 3^{2}\cdot 137\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4932}(2465,·)$, $\chi_{4932}(547,·)$, $\chi_{4932}(1,·)$, $\chi_{4932}(3563,·)$, $\chi_{4932}(4109,·)$, $\chi_{4932}(2191,·)$, $\chi_{4932}(1919,·)$, $\chi_{4932}(275,·)$, $\chi_{4932}(821,·)$, $\chi_{4932}(3289,·)$, $\chi_{4932}(3835,·)$, $\chi_{4932}(1645,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-411}) \), \(\Q(\sqrt{-137}) \), 6.0.50611941099.4$^{3}$, 6.0.1079721410112.8$^{3}$, 12.0.10492184911088211262672896.1$^{24}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19}a^{9}+\frac{7}{19}a^{7}-\frac{8}{19}a^{5}+\frac{3}{19}a^{3}+\frac{9}{19}a$, $\frac{1}{8352956312191}a^{10}+\frac{767521970142}{8352956312191}a^{8}+\frac{3025671332088}{8352956312191}a^{6}-\frac{1629466338595}{8352956312191}a^{4}+\frac{3131284356608}{8352956312191}a^{2}+\frac{43867932021}{439629279589}$, $\frac{1}{21\!\cdots\!57}a^{11}+\frac{346755765006685}{21\!\cdots\!57}a^{9}+\frac{66\!\cdots\!53}{21\!\cdots\!57}a^{7}-\frac{255295560661448}{21\!\cdots\!57}a^{5}-\frac{57\!\cdots\!92}{21\!\cdots\!57}a^{3}-\frac{10\!\cdots\!75}{21\!\cdots\!57}a$
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
$C_{3}\times C_{68328}$, which has order $204984$ (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: |
$\frac{35649310}{8352956312191}a^{10}+\frac{6772462758}{8352956312191}a^{8}+\frac{531560476956}{8352956312191}a^{6}+\frac{22737591049209}{8352956312191}a^{4}+\frac{529984889630467}{8352956312191}a^{2}+\frac{275132630758932}{439629279589}$, $\frac{35649310}{8352956312191}a^{10}+\frac{6772462758}{8352956312191}a^{8}+\frac{531560476956}{8352956312191}a^{6}+\frac{22737591049209}{8352956312191}a^{4}+\frac{529984889630467}{8352956312191}a^{2}+\frac{274693001479343}{439629279589}$, $\frac{128732167357}{21\!\cdots\!57}a^{11}-\frac{11973748}{8352956312191}a^{10}+\frac{19919709313320}{21\!\cdots\!57}a^{9}-\frac{2276113623}{8352956312191}a^{8}+\frac{13\!\cdots\!71}{21\!\cdots\!57}a^{7}-\frac{178972352422}{8352956312191}a^{6}+\frac{52\!\cdots\!69}{21\!\cdots\!57}a^{5}-\frac{7688240221549}{8352956312191}a^{4}+\frac{10\!\cdots\!49}{21\!\cdots\!57}a^{3}-\frac{180217203677330}{8352956312191}a^{2}+\frac{93\!\cdots\!10}{21\!\cdots\!57}a-\frac{94295894114211}{439629279589}$, $\frac{33385683259}{21\!\cdots\!57}a^{11}-\frac{35649310}{8352956312191}a^{10}+\frac{5206074227710}{21\!\cdots\!57}a^{9}-\frac{6772462758}{8352956312191}a^{8}+\frac{362177681549152}{21\!\cdots\!57}a^{7}-\frac{531560476956}{8352956312191}a^{6}+\frac{13\!\cdots\!86}{21\!\cdots\!57}a^{5}-\frac{22737591049209}{8352956312191}a^{4}+\frac{29\!\cdots\!83}{21\!\cdots\!57}a^{3}-\frac{529984889630467}{8352956312191}a^{2}+\frac{25\!\cdots\!73}{21\!\cdots\!57}a-\frac{274693001479343}{439629279589}$, $\frac{128732167357}{21\!\cdots\!57}a^{11}+\frac{19919709313320}{21\!\cdots\!57}a^{9}+\frac{13\!\cdots\!71}{21\!\cdots\!57}a^{7}+\frac{52\!\cdots\!69}{21\!\cdots\!57}a^{5}+\frac{10\!\cdots\!49}{21\!\cdots\!57}a^{3}+\frac{93\!\cdots\!10}{21\!\cdots\!57}a-1$
| 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
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 325.67540279491664 \cdot 204984}{2\cdot\sqrt{10492184911088211262672896}}\cr\approx \mathstrut & 0.634046586929141 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 + 189*x^10 + 15972*x^8 + 773671*x^6 + 22405545*x^4 + 361326234*x^2 + 2491307569)
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 + 189*x^10 + 15972*x^8 + 773671*x^6 + 22405545*x^4 + 361326234*x^2 + 2491307569, 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 + 189*x^10 + 15972*x^8 + 773671*x^6 + 22405545*x^4 + 361326234*x^2 + 2491307569);
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 + 189*x^10 + 15972*x^8 + 773671*x^6 + 22405545*x^4 + 361326234*x^2 + 2491307569);
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
| \(\Q(\sqrt{-137}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-411}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{3}, \sqrt{-137})\), 6.0.1079721410112.8, \(\Q(\zeta_{36})^+\), 6.0.50611941099.4 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.1.0.1}{1} }^{12}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
|
\(3\)
| 3.6.9.3 | $x^{6} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
| 3.6.9.3 | $x^{6} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
|
\(137\)
| 137.12.6.1 | $x^{12} + 60828 x^{11} + 1541686484 x^{10} + 20839520730232 x^{9} + 158453825296277278 x^{8} + 642567107793763822910 x^{7} + 1085753296305906315560456 x^{6} + 88692636654066038418688 x^{5} + 1163262987800321522305993 x^{4} + 126068624279763415475578086 x^{3} + 115852172162316468136702157 x^{2} + 6321093873407672157898092 x + 3321652024360950282106020$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |