Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 44*x^10 + 224*x^9 + 117*x^8 - 2771*x^7 + 6359*x^6 - 4427*x^5 - 1991*x^4 + 3249*x^3 - 434*x^2 - 280*x + 56)
gp: K = bnfinit(y^12 - 3*y^11 - 44*y^10 + 224*y^9 + 117*y^8 - 2771*y^7 + 6359*y^6 - 4427*y^5 - 1991*y^4 + 3249*y^3 - 434*y^2 - 280*y + 56, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 - 44*x^10 + 224*x^9 + 117*x^8 - 2771*x^7 + 6359*x^6 - 4427*x^5 - 1991*x^4 + 3249*x^3 - 434*x^2 - 280*x + 56);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 3*x^11 - 44*x^10 + 224*x^9 + 117*x^8 - 2771*x^7 + 6359*x^6 - 4427*x^5 - 1991*x^4 + 3249*x^3 - 434*x^2 - 280*x + 56)
\( x^{12} - 3 x^{11} - 44 x^{10} + 224 x^{9} + 117 x^{8} - 2771 x^{7} + 6359 x^{6} - 4427 x^{5} - 1991 x^{4} + \cdots + 56 \)
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: |
\(125049841902709607569\)
\(\medspace = 7^{8}\cdot 167^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(47.29\) | 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: | $7^{2/3}167^{1/2}\approx 47.28865141512203$ | ||
| Ramified primes: |
\(7\), \(167\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{12}a^{9}-\frac{1}{12}a^{7}-\frac{1}{6}a^{6}-\frac{1}{12}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{5}{12}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{12}a^{10}-\frac{1}{12}a^{8}+\frac{1}{12}a^{7}+\frac{1}{6}a^{6}+\frac{1}{4}a^{4}+\frac{5}{12}a^{3}-\frac{5}{12}a^{2}+\frac{1}{6}a$, $\frac{1}{70690452}a^{11}+\frac{402335}{23563484}a^{10}-\frac{725143}{70690452}a^{9}+\frac{6320389}{70690452}a^{8}-\frac{1555459}{17672613}a^{7}-\frac{5814327}{23563484}a^{6}+\frac{728955}{23563484}a^{5}+\frac{411506}{2524659}a^{4}-\frac{7813319}{70690452}a^{3}+\frac{3934007}{10098636}a^{2}+\frac{440001}{1683106}a+\frac{132469}{841553}$
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 $1$ (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: | $11$ | 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: |
$\frac{352249}{23563484}a^{11}+\frac{408236}{17672613}a^{10}-\frac{24713119}{35345226}a^{9}+\frac{17920009}{70690452}a^{8}+\frac{678081451}{70690452}a^{7}-\frac{509421341}{35345226}a^{6}-\frac{532176523}{17672613}a^{5}+\frac{108804461}{1683106}a^{4}+\frac{57107783}{35345226}a^{3}-\frac{397057409}{10098636}a^{2}+\frac{1987640}{2524659}a+\frac{10243181}{2524659}$, $\frac{521749}{23563484}a^{11}-\frac{3087023}{35345226}a^{10}-\frac{69134977}{70690452}a^{9}+\frac{418469041}{70690452}a^{8}+\frac{11003661}{11781742}a^{7}-\frac{420764284}{5890871}a^{6}+\frac{12181353397}{70690452}a^{5}-\frac{367951609}{3366212}a^{4}-\frac{5580585725}{70690452}a^{3}+\frac{72488992}{841553}a^{2}+\frac{8516637}{1683106}a-\frac{16507352}{2524659}$, $\frac{2543599}{70690452}a^{11}-\frac{219767}{5890871}a^{10}-\frac{29470220}{17672613}a^{9}+\frac{85572535}{17672613}a^{8}+\frac{83825586}{5890871}a^{7}-\frac{5319372337}{70690452}a^{6}+\frac{2851015337}{35345226}a^{5}+\frac{205099993}{5049318}a^{4}-\frac{6045408221}{70690452}a^{3}-\frac{6227551}{3366212}a^{2}+\frac{96778891}{5049318}a-\frac{7535429}{2524659}$, $\frac{714191}{70690452}a^{11}-\frac{869653}{23563484}a^{10}-\frac{15012787}{35345226}a^{9}+\frac{46295339}{17672613}a^{8}-\frac{6403591}{35345226}a^{7}-\frac{371248361}{11781742}a^{6}+\frac{987463347}{11781742}a^{5}-\frac{579787769}{10098636}a^{4}-\frac{3220280293}{70690452}a^{3}+\frac{230998325}{5049318}a^{2}+\frac{15596415}{1683106}a-\frac{1745340}{841553}$, $\frac{2777507}{70690452}a^{11}-\frac{197605}{35345226}a^{10}-\frac{31167973}{17672613}a^{9}+\frac{66150064}{17672613}a^{8}+\frac{284492914}{17672613}a^{7}-\frac{1502210687}{23563484}a^{6}+\frac{1059605168}{17672613}a^{5}+\frac{82709189}{5049318}a^{4}-\frac{3302482205}{70690452}a^{3}+\frac{149209453}{10098636}a^{2}+\frac{8448654}{841553}a-\frac{7120147}{2524659}$, $\frac{2687615}{10098636}a^{11}-\frac{6634357}{10098636}a^{10}-\frac{120921845}{10098636}a^{9}+\frac{44790251}{841553}a^{8}+\frac{93531737}{1683106}a^{7}-\frac{7060371623}{10098636}a^{6}+\frac{1141321349}{841553}a^{5}-\frac{1525565777}{2524659}a^{4}-\frac{606305571}{841553}a^{3}+\frac{918666703}{1683106}a^{2}+\frac{409972151}{5049318}a-\frac{34065593}{841553}$, $\frac{999949}{10098636}a^{11}-\frac{217731}{841553}a^{10}-\frac{3749311}{841553}a^{9}+\frac{103222217}{5049318}a^{8}+\frac{196252291}{10098636}a^{7}-\frac{1343525335}{5049318}a^{6}+\frac{5310079193}{10098636}a^{5}-\frac{1231477007}{5049318}a^{4}-\frac{2670783557}{10098636}a^{3}+\frac{523320235}{2524659}a^{2}+\frac{83829431}{5049318}a-\frac{47157352}{2524659}$, $\frac{126057}{1683106}a^{11}-\frac{2227819}{10098636}a^{10}-\frac{33689419}{10098636}a^{9}+\frac{41948068}{2524659}a^{8}+\frac{36939825}{3366212}a^{7}-\frac{708101107}{3366212}a^{6}+\frac{4581541807}{10098636}a^{5}-\frac{426523343}{1683106}a^{4}-\frac{2249095409}{10098636}a^{3}+\frac{190509476}{841553}a^{2}+\frac{24854905}{1683106}a-\frac{70384316}{2524659}$, $\frac{2953051}{70690452}a^{11}-\frac{378809}{35345226}a^{10}-\frac{32308529}{17672613}a^{9}+\frac{76971554}{17672613}a^{8}+\frac{532065523}{35345226}a^{7}-\frac{1685064503}{23563484}a^{6}+\frac{3123923393}{35345226}a^{5}-\frac{3485641}{2524659}a^{4}-\frac{4914574723}{70690452}a^{3}+\frac{296790941}{10098636}a^{2}+\frac{12715468}{841553}a-\frac{15340310}{2524659}$, $\frac{3696911}{35345226}a^{11}-\frac{12570953}{70690452}a^{10}-\frac{27659556}{5890871}a^{9}+\frac{204100187}{11781742}a^{8}+\frac{1010769139}{35345226}a^{7}-\frac{5656174575}{23563484}a^{6}+\frac{7255673573}{17672613}a^{5}-\frac{1515376957}{10098636}a^{4}-\frac{2553130385}{11781742}a^{3}+\frac{1551482657}{10098636}a^{2}+\frac{29475929}{1683106}a-\frac{37606495}{2524659}$, $\frac{177425}{17672613}a^{11}-\frac{9557965}{70690452}a^{10}-\frac{3213047}{11781742}a^{9}+\frac{82457833}{11781742}a^{8}-\frac{1106644865}{70690452}a^{7}-\frac{345199844}{5890871}a^{6}+\frac{21069080665}{70690452}a^{5}-\frac{4296178049}{10098636}a^{4}+\frac{1508016847}{11781742}a^{3}+\frac{843197255}{5049318}a^{2}-\frac{93227176}{841553}a+\frac{44594683}{2524659}$
| 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: | \( 2838054.22304 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 2838054.22304 \cdot 1}{2\cdot\sqrt{125049841902709607569}}\cr\approx \mathstrut & 0.519767436619 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 44*x^10 + 224*x^9 + 117*x^8 - 2771*x^7 + 6359*x^6 - 4427*x^5 - 1991*x^4 + 3249*x^3 - 434*x^2 - 280*x + 56)
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 - 3*x^11 - 44*x^10 + 224*x^9 + 117*x^8 - 2771*x^7 + 6359*x^6 - 4427*x^5 - 1991*x^4 + 3249*x^3 - 434*x^2 - 280*x + 56, 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 - 3*x^11 - 44*x^10 + 224*x^9 + 117*x^8 - 2771*x^7 + 6359*x^6 - 4427*x^5 - 1991*x^4 + 3249*x^3 - 434*x^2 - 280*x + 56);
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 - 3*x^11 - 44*x^10 + 224*x^9 + 117*x^8 - 2771*x^7 + 6359*x^6 - 4427*x^5 - 1991*x^4 + 3249*x^3 - 434*x^2 - 280*x + 56);
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
| \(\Q(\zeta_{7})^+\), 4.4.1366561.1 x4, 6.6.66961489.1 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: | 4.4.1366561.1 |
| Degree 6 sibling: | 6.6.66961489.1 |
| Minimal sibling: | 4.4.1366561.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.1.0.1}{1} }^{12}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
|
\(167\)
| 167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |