Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 714*x^10 + 197064*x^8 - 26655048*x^6 + 1835265600*x^4 - 59903069184*x^2 + 718836830208)
gp: K = bnfinit(y^12 - 714*y^10 + 197064*y^8 - 26655048*y^6 + 1835265600*y^4 - 59903069184*y^2 + 718836830208, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 714*x^10 + 197064*x^8 - 26655048*x^6 + 1835265600*x^4 - 59903069184*x^2 + 718836830208);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 714*x^10 + 197064*x^8 - 26655048*x^6 + 1835265600*x^4 - 59903069184*x^2 + 718836830208)
\( x^{12} - 714x^{10} + 197064x^{8} - 26655048x^{6} + 1835265600x^{4} - 59903069184x^{2} + 718836830208 \)
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: |
\(6401594233356076787154812928\)
\(\medspace = 2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(207.58\) | 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: | $2^{3/2}3^{1/2}7^{5/6}17^{3/4}\approx 207.58247157811073$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2856=2^{3}\cdot 3\cdot 7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2856}(1,·)$, $\chi_{2856}(965,·)$, $\chi_{2856}(1781,·)$, $\chi_{2856}(2209,·)$, $\chi_{2856}(1801,·)$, $\chi_{2856}(2189,·)$, $\chi_{2856}(1517,·)$, $\chi_{2856}(2041,·)$, $\chi_{2856}(1109,·)$, $\chi_{2856}(1633,·)$, $\chi_{2856}(169,·)$, $\chi_{2856}(293,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{6}a^{2}$, $\frac{1}{6}a^{3}$, $\frac{1}{612}a^{4}$, $\frac{1}{612}a^{5}$, $\frac{1}{25704}a^{6}$, $\frac{1}{51408}a^{7}-\frac{1}{1224}a^{5}-\frac{1}{2}a$, $\frac{1}{10487232}a^{8}+\frac{1}{102816}a^{6}-\frac{1}{1224}a^{4}-\frac{1}{24}a^{2}$, $\frac{1}{20974464}a^{9}+\frac{1}{205632}a^{7}-\frac{1}{2448}a^{5}+\frac{1}{16}a^{3}-\frac{1}{2}a$, $\frac{1}{3272016384}a^{10}-\frac{1}{77905152}a^{8}-\frac{5}{891072}a^{6}-\frac{61}{127296}a^{4}-\frac{5}{312}a^{2}-\frac{1}{13}$, $\frac{1}{6544032768}a^{11}-\frac{1}{155810304}a^{9}-\frac{5}{1782144}a^{7}-\frac{61}{254592}a^{5}-\frac{5}{624}a^{3}-\frac{1}{26}a$
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
$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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: |
$\frac{1}{409002048}a^{10}-\frac{61}{45444672}a^{8}+\frac{49}{190944}a^{6}-\frac{167}{7956}a^{4}+\frac{233}{312}a^{2}-\frac{151}{13}$, $\frac{5}{1090672128}a^{10}-\frac{1457}{545336064}a^{8}+\frac{71}{127296}a^{6}-\frac{2177}{42432}a^{4}+\frac{209}{104}a^{2}-\frac{327}{13}$, $\frac{23}{3272016384}a^{10}-\frac{695}{181778688}a^{8}+\frac{71}{99008}a^{6}-\frac{419}{7488}a^{4}+\frac{87}{52}a^{2}-\frac{153}{13}$, $\frac{1}{3272016384}a^{10}-\frac{215}{545336064}a^{8}+\frac{401}{2673216}a^{6}-\frac{991}{42432}a^{4}+\frac{463}{312}a^{2}-\frac{352}{13}$, $\frac{1}{1636008192}a^{10}-\frac{5}{16039296}a^{8}+\frac{25}{668304}a^{6}+\frac{199}{63648}a^{4}-\frac{77}{104}a^{2}+\frac{336}{13}$, $\frac{59}{6544032768}a^{11}-\frac{1}{25562628}a^{10}-\frac{839}{155810304}a^{9}+\frac{5}{250614}a^{8}+\frac{895}{763776}a^{7}-\frac{139}{41769}a^{6}-\frac{1735}{14976}a^{5}+\frac{8}{39}a^{4}+\frac{1601}{312}a^{3}-\frac{128}{39}a^{2}-\frac{2009}{26}a-\frac{67}{13}$, $\frac{31}{6544032768}a^{11}-\frac{1}{25562628}a^{10}-\frac{991}{363557376}a^{9}+\frac{5}{250614}a^{8}+\frac{145}{254592}a^{7}-\frac{139}{41769}a^{6}-\frac{827}{14976}a^{5}+\frac{8}{39}a^{4}+\frac{271}{104}a^{3}-\frac{128}{39}a^{2}-\frac{1253}{26}a-\frac{613}{13}$, $\frac{737}{3272016384}a^{11}-\frac{8773}{3272016384}a^{10}-\frac{3331}{25968384}a^{9}+\frac{834079}{545336064}a^{8}+\frac{22991}{891072}a^{7}-\frac{48439}{157248}a^{6}-\frac{287693}{127296}a^{5}+\frac{202217}{7488}a^{4}+\frac{26605}{312}a^{3}-\frac{158995}{156}a^{2}-\frac{14517}{13}a+\frac{173457}{13}$, $\frac{83}{818004096}a^{11}+\frac{6073}{3272016384}a^{10}-\frac{15}{210392}a^{9}-\frac{683983}{545336064}a^{8}+\frac{25577}{1336608}a^{7}+\frac{850937}{2673216}a^{6}-\frac{78071}{31824}a^{5}-\frac{4785461}{127296}a^{4}+\frac{15597}{104}a^{3}+\frac{632063}{312}a^{2}-\frac{90793}{26}a-\frac{498162}{13}$, $\frac{4909}{6544032768}a^{11}+\frac{14831}{1636008192}a^{10}-\frac{17177}{40395264}a^{9}-\frac{200393}{38952576}a^{8}+\frac{455881}{5346432}a^{7}+\frac{172517}{167076}a^{6}-\frac{1897097}{254592}a^{5}-\frac{338119}{3744}a^{4}+\frac{175057}{624}a^{3}+\frac{1061561}{312}a^{2}-\frac{95311}{26}a-\frac{578249}{13}$, $\frac{415}{818004096}a^{11}-\frac{2381}{363557376}a^{10}-\frac{6319}{22722336}a^{9}+\frac{93671}{25968384}a^{8}+\frac{71465}{1336608}a^{7}-\frac{622103}{891072}a^{6}-\frac{142445}{31824}a^{5}+\frac{7498705}{127296}a^{4}+\frac{50941}{312}a^{3}-\frac{224805}{104}a^{2}-\frac{27179}{13}a+\frac{361106}{13}$
| 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: | \( 1441606350.4875915 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 1441606350.4875915 \cdot 16}{2\cdot\sqrt{6401594233356076787154812928}}\cr\approx \mathstrut & 0.590408430674939 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 714*x^10 + 197064*x^8 - 26655048*x^6 + 1835265600*x^4 - 59903069184*x^2 + 718836830208)
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 - 714*x^10 + 197064*x^8 - 26655048*x^6 + 1835265600*x^4 - 59903069184*x^2 + 718836830208, 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 - 714*x^10 + 197064*x^8 - 26655048*x^6 + 1835265600*x^4 - 59903069184*x^2 + 718836830208);
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 - 714*x^10 + 197064*x^8 - 26655048*x^6 + 1835265600*x^4 - 59903069184*x^2 + 718836830208);
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 cyclic group of order 12 |
| The 12 conjugacy class representatives for $C_{12}$ |
| Character table for $C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), 4.4.138664512.1, 6.6.11796113.1 |
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]
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 | ${\href{/padicField/5.12.0.1}{12} }$ | R | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.1.0.1}{1} }^{12}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.1.0.1}{1} }^{12}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.9.1 | $x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ |
| 2.6.9.1 | $x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
|
\(3\)
| 3.12.6.1 | $x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |
|
\(7\)
| 7.12.10.5 | $x^{12} - 154 x^{6} - 1421$ | $6$ | $2$ | $10$ | $C_{12}$ | $[\ ]_{6}^{2}$ |
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |