Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 7*x^10 + 6*x^9 + 22*x^8 + 62*x^7 - 157*x^6 - 152*x^5 + 282*x^4 + 200*x^3 - 115*x^2 - 72*x - 9)
gp: K = bnfinit(y^12 - 2*y^11 - 7*y^10 + 6*y^9 + 22*y^8 + 62*y^7 - 157*y^6 - 152*y^5 + 282*y^4 + 200*y^3 - 115*y^2 - 72*y - 9, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 - 7*x^10 + 6*x^9 + 22*x^8 + 62*x^7 - 157*x^6 - 152*x^5 + 282*x^4 + 200*x^3 - 115*x^2 - 72*x - 9);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 - 7*x^10 + 6*x^9 + 22*x^8 + 62*x^7 - 157*x^6 - 152*x^5 + 282*x^4 + 200*x^3 - 115*x^2 - 72*x - 9)
\( x^{12} - 2 x^{11} - 7 x^{10} + 6 x^{9} + 22 x^{8} + 62 x^{7} - 157 x^{6} - 152 x^{5} + 282 x^{4} + \cdots - 9 \)
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: |
\(14963642447137024\)
\(\medspace = 2^{8}\cdot 197^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.28\) | 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: | $2^{2/3}197^{1/2}\approx 22.280235493786417$ | ||
| Ramified primes: |
\(2\), \(197\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}$, $\frac{1}{4}a^{9}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}+\frac{1}{8}a^{6}+\frac{1}{8}a^{5}+\frac{3}{8}a^{4}+\frac{1}{4}a^{3}-\frac{3}{8}a^{2}+\frac{3}{8}a+\frac{3}{8}$, $\frac{1}{80591640}a^{11}-\frac{1398079}{40295820}a^{10}+\frac{3413773}{40295820}a^{9}-\frac{239297}{5372776}a^{8}-\frac{15305453}{80591640}a^{7}-\frac{165755}{4029582}a^{6}+\frac{4678996}{10073955}a^{5}-\frac{3990139}{16118328}a^{4}+\frac{11537769}{26863880}a^{3}+\frac{3500329}{40295820}a^{2}+\frac{895351}{40295820}a+\frac{1977977}{26863880}$
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: |
$\frac{16311649}{80591640}a^{11}-\frac{17995201}{40295820}a^{10}-\frac{26620219}{20147910}a^{9}+\frac{7868381}{5372776}a^{8}+\frac{330594763}{80591640}a^{7}+\frac{47668693}{4029582}a^{6}-\frac{1365185099}{40295820}a^{5}-\frac{377589649}{16118328}a^{4}+\frac{1621094741}{26863880}a^{3}+\frac{1051118251}{40295820}a^{2}-\frac{255510794}{10073955}a-\frac{158502777}{26863880}$, $\frac{17540081}{80591640}a^{11}-\frac{39740173}{80591640}a^{10}-\frac{112541759}{80591640}a^{9}+\frac{4570481}{2686388}a^{8}+\frac{349748417}{80591640}a^{7}+\frac{196966673}{16118328}a^{6}-\frac{3023241947}{80591640}a^{5}-\frac{187112725}{8059164}a^{4}+\frac{1864577779}{26863880}a^{3}+\frac{1950270583}{80591640}a^{2}-\frac{2732363783}{80591640}a-\frac{42354197}{6715970}$, $\frac{2616773}{8059164}a^{11}-\frac{5836489}{8059164}a^{10}-\frac{4283465}{2014791}a^{9}+\frac{3272257}{1343194}a^{8}+\frac{54197855}{8059164}a^{7}+\frac{150918715}{8059164}a^{6}-\frac{111620615}{2014791}a^{5}-\frac{152956493}{4029582}a^{4}+\frac{269997985}{2686388}a^{3}+\frac{364949521}{8059164}a^{2}-\frac{96516914}{2014791}a-\frac{17465293}{1343194}$, $\frac{31095059}{40295820}a^{11}-\frac{139383659}{80591640}a^{10}-\frac{401766307}{80591640}a^{9}+\frac{31418655}{5372776}a^{8}+\frac{156851747}{10073955}a^{7}+\frac{709348417}{16118328}a^{6}-\frac{10628498161}{80591640}a^{5}-\frac{1379024659}{16118328}a^{4}+\frac{1606854603}{6715970}a^{3}+\frac{7799762969}{80591640}a^{2}-\frac{9174739279}{80591640}a-\frac{785987109}{26863880}$, $\frac{1425925}{16118328}a^{11}-\frac{3405221}{16118328}a^{10}-\frac{8365873}{16118328}a^{9}+\frac{454910}{671597}a^{8}+\frac{25788313}{16118328}a^{7}+\frac{81135221}{16118328}a^{6}-\frac{251169445}{16118328}a^{5}-\frac{25348625}{4029582}a^{4}+\frac{127356827}{5372776}a^{3}+\frac{137293523}{16118328}a^{2}-\frac{149438005}{16118328}a-\frac{6363491}{2686388}$, $\frac{2057825}{16118328}a^{11}-\frac{4530061}{16118328}a^{10}-\frac{14460503}{16118328}a^{9}+\frac{2918041}{2686388}a^{8}+\frac{46801601}{16118328}a^{7}+\frac{112121233}{16118328}a^{6}-\frac{358889087}{16118328}a^{5}-\frac{148769225}{8059164}a^{4}+\frac{265940279}{5372776}a^{3}+\frac{282733255}{16118328}a^{2}-\frac{470102603}{16118328}a-\frac{3604378}{671597}$, $\frac{31095059}{40295820}a^{11}-\frac{139383659}{80591640}a^{10}-\frac{401766307}{80591640}a^{9}+\frac{31418655}{5372776}a^{8}+\frac{156851747}{10073955}a^{7}+\frac{709348417}{16118328}a^{6}-\frac{10628498161}{80591640}a^{5}-\frac{1379024659}{16118328}a^{4}+\frac{1606854603}{6715970}a^{3}+\frac{7799762969}{80591640}a^{2}-\frac{9094147639}{80591640}a-\frac{759123229}{26863880}$
| 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: | \( 4040.08904277 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 4040.08904277 \cdot 1}{2\cdot\sqrt{14963642447137024}}\cr\approx \mathstrut & 0.411795632577 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 7*x^10 + 6*x^9 + 22*x^8 + 62*x^7 - 157*x^6 - 152*x^5 + 282*x^4 + 200*x^3 - 115*x^2 - 72*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 - 2*x^11 - 7*x^10 + 6*x^9 + 22*x^8 + 62*x^7 - 157*x^6 - 152*x^5 + 282*x^4 + 200*x^3 - 115*x^2 - 72*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 - 2*x^11 - 7*x^10 + 6*x^9 + 22*x^8 + 62*x^7 - 157*x^6 - 152*x^5 + 282*x^4 + 200*x^3 - 115*x^2 - 72*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 - 2*x^11 - 7*x^10 + 6*x^9 + 22*x^8 + 62*x^7 - 157*x^6 - 152*x^5 + 282*x^4 + 200*x^3 - 115*x^2 - 72*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
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
| \(\Q(\sqrt{197}) \), 3.3.788.1 x3, 6.2.122325968.1, 6.2.620944.1, 6.6.122325968.1 |
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: | 4.0.788.1 |
| Degree 6 siblings: | 6.2.122325968.1, 6.2.620944.1 |
| Degree 8 sibling: | 8.0.24098215696.1 |
| Degree 12 sibling: | 12.0.75957575873792.1 |
| Minimal sibling: | 4.0.788.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.1.0.1}{1} }^{12}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(197\)
| 197.2.1.1 | $x^{2} + 197$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 197.2.1.1 | $x^{2} + 197$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 197.4.2.1 | $x^{4} + 66970 x^{3} + 1127637879 x^{2} + 214058019190 x + 3496206875$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 197.4.2.1 | $x^{4} + 66970 x^{3} + 1127637879 x^{2} + 214058019190 x + 3496206875$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |