Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 27*x^10 + 70*x^9 + 183*x^8 - 604*x^7 - 300*x^6 + 1896*x^5 - 467*x^4 - 2170*x^3 + 1207*x^2 + 602*x - 373)
gp: K = bnfinit(y^12 - 2*y^11 - 27*y^10 + 70*y^9 + 183*y^8 - 604*y^7 - 300*y^6 + 1896*y^5 - 467*y^4 - 2170*y^3 + 1207*y^2 + 602*y - 373, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 - 27*x^10 + 70*x^9 + 183*x^8 - 604*x^7 - 300*x^6 + 1896*x^5 - 467*x^4 - 2170*x^3 + 1207*x^2 + 602*x - 373);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 - 27*x^10 + 70*x^9 + 183*x^8 - 604*x^7 - 300*x^6 + 1896*x^5 - 467*x^4 - 2170*x^3 + 1207*x^2 + 602*x - 373)
\( x^{12} - 2 x^{11} - 27 x^{10} + 70 x^{9} + 183 x^{8} - 604 x^{7} - 300 x^{6} + 1896 x^{5} - 467 x^{4} + \cdots - 373 \)
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: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(223580268118933504\)
\(\medspace = 2^{18}\cdot 31^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(27.91\) | 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: | $2^{3/2}31^{2/3}\approx 27.911689339648817$ | ||
| Ramified primes: |
\(2\), \(31\)
| 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 over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{48}a^{10}-\frac{1}{12}a^{9}-\frac{1}{24}a^{8}+\frac{1}{8}a^{7}-\frac{7}{48}a^{6}-\frac{1}{6}a^{5}+\frac{5}{48}a^{4}-\frac{1}{24}a^{3}-\frac{3}{8}a^{2}-\frac{1}{6}a-\frac{19}{48}$, $\frac{1}{39888}a^{11}-\frac{37}{13296}a^{10}-\frac{481}{6648}a^{9}-\frac{1115}{9972}a^{8}-\frac{2305}{39888}a^{7}+\frac{149}{4432}a^{6}-\frac{625}{13296}a^{5}+\frac{759}{4432}a^{4}-\frac{533}{9972}a^{3}-\frac{9541}{19944}a^{2}-\frac{1001}{13296}a-\frac{11119}{39888}$
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: | $11$ | 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: |
$\frac{17579}{19944}a^{11}-\frac{6979}{13296}a^{10}-\frac{20378}{831}a^{9}+\frac{543491}{19944}a^{8}+\frac{496622}{2493}a^{7}-\frac{1117777}{4432}a^{6}-\frac{4092899}{6648}a^{5}+\frac{3559023}{4432}a^{4}+\frac{14094263}{19944}a^{3}-\frac{18155875}{19944}a^{2}-\frac{1348867}{6648}a+\frac{9476435}{39888}$, $\frac{17579}{19944}a^{11}-\frac{6979}{13296}a^{10}-\frac{20378}{831}a^{9}+\frac{543491}{19944}a^{8}+\frac{496622}{2493}a^{7}-\frac{1117777}{4432}a^{6}-\frac{4092899}{6648}a^{5}+\frac{3559023}{4432}a^{4}+\frac{14094263}{19944}a^{3}-\frac{18155875}{19944}a^{2}-\frac{1348867}{6648}a+\frac{9516323}{39888}$, $\frac{5911}{1108}a^{11}-\frac{42385}{13296}a^{10}-\frac{123410}{831}a^{9}+\frac{1098025}{6648}a^{8}+\frac{2676539}{2216}a^{7}-\frac{20320769}{13296}a^{6}-\frac{12442243}{3324}a^{5}+\frac{64687315}{13296}a^{4}+\frac{28778647}{6648}a^{3}-\frac{12199069}{2216}a^{2}-\frac{2119217}{1662}a+\frac{18898099}{13296}$, $\frac{100733}{39888}a^{11}-\frac{10015}{6648}a^{10}-\frac{467309}{6648}a^{9}+\frac{1557793}{19944}a^{8}+\frac{22799461}{39888}a^{7}-\frac{1601351}{2216}a^{6}-\frac{23542133}{13296}a^{5}+\frac{636835}{277}a^{4}+\frac{40805035}{19944}a^{3}-\frac{25909585}{9972}a^{2}-\frac{8017681}{13296}a+\frac{13366133}{19944}$, $\frac{79123}{39888}a^{11}-\frac{7171}{6648}a^{10}-\frac{366773}{6648}a^{9}+\frac{1167833}{19944}a^{8}+\frac{17924063}{39888}a^{7}-\frac{3626035}{6648}a^{6}-\frac{18569699}{13296}a^{5}+\frac{5784397}{3324}a^{4}+\frac{32343071}{19944}a^{3}-\frac{9836065}{4986}a^{2}-\frac{2123689}{4432}a+\frac{10167415}{19944}$, $\frac{53077}{19944}a^{11}-\frac{21721}{13296}a^{10}-\frac{61577}{831}a^{9}+\frac{1666381}{19944}a^{8}+\frac{3003191}{4986}a^{7}-\frac{10259065}{13296}a^{6}-\frac{12395909}{6648}a^{5}+\frac{32631119}{13296}a^{4}+\frac{42930721}{19944}a^{3}-\frac{55321325}{19944}a^{2}-\frac{1400635}{2216}a+\frac{28473697}{39888}$, $\frac{83221}{13296}a^{11}-\frac{12845}{3324}a^{10}-\frac{1159061}{6648}a^{9}+\frac{1310455}{6648}a^{8}+\frac{18855701}{13296}a^{7}-\frac{1513099}{831}a^{6}-\frac{58442303}{13296}a^{5}+\frac{38537755}{6648}a^{4}+\frac{33817087}{6648}a^{3}-\frac{5452424}{831}a^{2}-\frac{20020391}{13296}a+\frac{1406086}{831}$, $\frac{79123}{39888}a^{11}-\frac{7171}{6648}a^{10}-\frac{366773}{6648}a^{9}+\frac{1167833}{19944}a^{8}+\frac{17924063}{39888}a^{7}-\frac{3626035}{6648}a^{6}-\frac{18569699}{13296}a^{5}+\frac{5784397}{3324}a^{4}+\frac{32343071}{19944}a^{3}-\frac{9836065}{4986}a^{2}-\frac{2123689}{4432}a+\frac{10187359}{19944}$, $\frac{287231}{39888}a^{11}-\frac{9861}{2216}a^{10}-\frac{444469}{2216}a^{9}+\frac{4524493}{19944}a^{8}+\frac{65065855}{39888}a^{7}-\frac{13926043}{6648}a^{6}-\frac{22401949}{4432}a^{5}+\frac{22157371}{3324}a^{4}+\frac{116635357}{19944}a^{3}-\frac{18800752}{2493}a^{2}-\frac{23008715}{13296}a+\frac{38767055}{19944}$, $\frac{224947}{19944}a^{11}-\frac{23555}{3324}a^{10}-\frac{522275}{1662}a^{9}+\frac{893660}{2493}a^{8}+\frac{50968169}{19944}a^{7}-\frac{1831983}{554}a^{6}-\frac{52638835}{6648}a^{5}+\frac{2914664}{277}a^{4}+\frac{22837922}{2493}a^{3}-\frac{118712287}{9972}a^{2}-\frac{18025253}{6648}a+\frac{15288029}{4986}$, $\frac{22319}{19944}a^{11}-\frac{8719}{13296}a^{10}-\frac{103625}{3324}a^{9}+\frac{683339}{19944}a^{8}+\frac{2534261}{9972}a^{7}-\frac{1406037}{4432}a^{6}-\frac{5263763}{6648}a^{5}+\frac{4468659}{4432}a^{4}+\frac{18512801}{19944}a^{3}-\frac{22615999}{19944}a^{2}-\frac{1910353}{6648}a+\frac{11337119}{39888}$
| 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: | \( 86712.0400607 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 86712.0400607 \cdot 1}{2\cdot\sqrt{223580268118933504}}\cr\approx \mathstrut & 0.375571492411 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 27*x^10 + 70*x^9 + 183*x^8 - 604*x^7 - 300*x^6 + 1896*x^5 - 467*x^4 - 2170*x^3 + 1207*x^2 + 602*x - 373)
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 - 27*x^10 + 70*x^9 + 183*x^8 - 604*x^7 - 300*x^6 + 1896*x^5 - 467*x^4 - 2170*x^3 + 1207*x^2 + 602*x - 373, 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 - 27*x^10 + 70*x^9 + 183*x^8 - 604*x^7 - 300*x^6 + 1896*x^5 - 467*x^4 - 2170*x^3 + 1207*x^2 + 602*x - 373);
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 - 27*x^10 + 70*x^9 + 183*x^8 - 604*x^7 - 300*x^6 + 1896*x^5 - 467*x^4 - 2170*x^3 + 1207*x^2 + 602*x - 373);
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 12 |
| The 4 conjugacy class representatives for $A_4$ |
| Character table for $A_4$ |
Intermediate fields
| 3.3.961.1, 4.4.61504.1 x4, 6.6.59105344.1 x3 |
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 4 sibling: | 4.4.61504.1 |
| Degree 6 sibling: | 6.6.59105344.1 |
| Minimal sibling: | 4.4.61504.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
|
\(31\)
| 31.3.2.1 | $x^{3} + 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 31.3.2.1 | $x^{3} + 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 31.3.2.1 | $x^{3} + 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 31.3.2.1 | $x^{3} + 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |