Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 33*x^10 - 94*x^9 + 250*x^8 - 374*x^7 + 577*x^6 + 16*x^5 - 464*x^4 + 2376*x^3 - 445*x^2 + 620*x + 3844)
gp: K = bnfinit(y^12 - 6*y^11 + 33*y^10 - 94*y^9 + 250*y^8 - 374*y^7 + 577*y^6 + 16*y^5 - 464*y^4 + 2376*y^3 - 445*y^2 + 620*y + 3844, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 + 33*x^10 - 94*x^9 + 250*x^8 - 374*x^7 + 577*x^6 + 16*x^5 - 464*x^4 + 2376*x^3 - 445*x^2 + 620*x + 3844);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 6*x^11 + 33*x^10 - 94*x^9 + 250*x^8 - 374*x^7 + 577*x^6 + 16*x^5 - 464*x^4 + 2376*x^3 - 445*x^2 + 620*x + 3844)
\( x^{12} - 6 x^{11} + 33 x^{10} - 94 x^{9} + 250 x^{8} - 374 x^{7} + 577 x^{6} + 16 x^{5} - 464 x^{4} + \cdots + 3844 \)
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: |
\(11057373810786304\)
\(\medspace = 2^{18}\cdot 59^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.73\) | 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: | $2^{3/2}59^{1/2}\approx 21.72556098240043$ | ||
| Ramified primes: |
\(2\), \(59\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{43}a^{8}-\frac{4}{43}a^{7}+\frac{18}{43}a^{6}+\frac{2}{43}a^{5}+\frac{2}{43}a^{4}+\frac{12}{43}a^{3}-\frac{1}{43}a^{2}-\frac{2}{43}a+\frac{13}{43}$, $\frac{1}{43}a^{9}+\frac{2}{43}a^{7}-\frac{12}{43}a^{6}+\frac{10}{43}a^{5}+\frac{20}{43}a^{4}+\frac{4}{43}a^{3}-\frac{6}{43}a^{2}+\frac{5}{43}a+\frac{9}{43}$, $\frac{1}{46397}a^{10}+\frac{491}{46397}a^{9}-\frac{293}{46397}a^{8}+\frac{248}{1079}a^{7}-\frac{19749}{46397}a^{6}+\frac{15434}{46397}a^{5}-\frac{3408}{46397}a^{4}+\frac{13554}{46397}a^{3}+\frac{10555}{46397}a^{2}-\frac{6793}{46397}a-\frac{8274}{46397}$, $\frac{1}{399849346}a^{11}-\frac{778}{199924673}a^{10}-\frac{4224027}{399849346}a^{9}-\frac{1374246}{199924673}a^{8}-\frac{6354740}{15378821}a^{7}-\frac{48299458}{199924673}a^{6}+\frac{104276269}{399849346}a^{5}+\frac{89973478}{199924673}a^{4}+\frac{97967332}{199924673}a^{3}+\frac{86618629}{199924673}a^{2}-\frac{156522421}{399849346}a+\frac{1202995}{6449183}$
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: |
$\frac{109873}{199924673}a^{11}-\frac{119874}{15378821}a^{10}+\frac{6400247}{199924673}a^{9}-\frac{27331903}{199924673}a^{8}+\frac{48090194}{199924673}a^{7}-\frac{120627745}{199924673}a^{6}+\frac{71976715}{199924673}a^{5}-\frac{152558854}{199924673}a^{4}-\frac{368754260}{199924673}a^{3}+\frac{2049714}{199924673}a^{2}-\frac{518364181}{199924673}a-\frac{25527395}{6449183}$, $\frac{990909}{399849346}a^{11}-\frac{2589412}{199924673}a^{10}+\frac{25772551}{399849346}a^{9}-\frac{31727784}{199924673}a^{8}+\frac{70130147}{199924673}a^{7}-\frac{90233027}{199924673}a^{6}+\frac{151569953}{399849346}a^{5}+\frac{94857410}{199924673}a^{4}-\frac{361692441}{199924673}a^{3}+\frac{225637408}{199924673}a^{2}-\frac{65194639}{399849346}a-\frac{24874896}{6449183}$, $\frac{428873}{199924673}a^{11}-\frac{1284258}{199924673}a^{10}+\frac{7837148}{199924673}a^{9}-\frac{8052517}{199924673}a^{8}+\frac{2452897}{15378821}a^{7}+\frac{13333248}{199924673}a^{6}+\frac{37816641}{199924673}a^{5}+\frac{201963692}{199924673}a^{4}+\frac{98439961}{199924673}a^{3}+\frac{215512008}{199924673}a^{2}+\frac{468536085}{199924673}a+\frac{3804885}{6449183}$, $\frac{3110}{15378821}a^{11}-\frac{885334}{199924673}a^{10}+\frac{4770478}{199924673}a^{9}-\frac{22265694}{199924673}a^{8}+\frac{57030123}{199924673}a^{7}-\frac{134099781}{199924673}a^{6}+\frac{168930491}{199924673}a^{5}-\frac{190149905}{199924673}a^{4}-\frac{213421168}{199924673}a^{3}+\frac{519081883}{199924673}a^{2}-\frac{1006358739}{199924673}a+\frac{2013369}{6449183}$, $\frac{181143}{4649411}a^{11}+\frac{21303534}{199924673}a^{10}-\frac{38575850}{199924673}a^{9}+\frac{806822589}{199924673}a^{8}-\frac{1101342650}{199924673}a^{7}+\frac{4566338014}{199924673}a^{6}-\frac{1495144894}{199924673}a^{5}+\frac{8672709914}{199924673}a^{4}+\frac{18850327290}{199924673}a^{3}-\frac{3614925983}{199924673}a^{2}+\frac{2625188873}{15378821}a+\frac{1542832541}{6449183}$
| 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: | \( 1021.43464739 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1021.43464739 \cdot 4}{2\cdot\sqrt{11057373810786304}}\cr\approx \mathstrut & 1.19534644723 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 33*x^10 - 94*x^9 + 250*x^8 - 374*x^7 + 577*x^6 + 16*x^5 - 464*x^4 + 2376*x^3 - 445*x^2 + 620*x + 3844)
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 - 6*x^11 + 33*x^10 - 94*x^9 + 250*x^8 - 374*x^7 + 577*x^6 + 16*x^5 - 464*x^4 + 2376*x^3 - 445*x^2 + 620*x + 3844, 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 - 6*x^11 + 33*x^10 - 94*x^9 + 250*x^8 - 374*x^7 + 577*x^6 + 16*x^5 - 464*x^4 + 2376*x^3 - 445*x^2 + 620*x + 3844);
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 - 6*x^11 + 33*x^10 - 94*x^9 + 250*x^8 - 374*x^7 + 577*x^6 + 16*x^5 - 464*x^4 + 2376*x^3 - 445*x^2 + 620*x + 3844);
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
$C_2\times S_4$ (as 12T24):
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 48 |
| The 10 conjugacy class representatives for $C_2 \times S_4$ |
| Character table for $C_2 \times S_4$ |
Intermediate fields
| \(\Q(\sqrt{118}) \), 3.1.59.1, 6.2.105154048.1, 6.0.27848.1, 6.0.13144256.3 |
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 6 siblings: | 6.2.1643032.1, 6.0.27848.1 |
| Degree 8 siblings: | 8.0.14258176.2, 8.4.49632710656.3 |
| Degree 12 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Minimal sibling: | 6.0.27848.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | R |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.8.12.2 | $x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
|
\(59\)
| 59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 59.4.2.1 | $x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 59.4.2.1 | $x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |