Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 65*x^10 + 1183*x^8 - 7670*x^6 + 14651*x^4 - 3809*x^2 + 13)
gp: K = bnfinit(y^12 - 65*y^10 + 1183*y^8 - 7670*y^6 + 14651*y^4 - 3809*y^2 + 13, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 65*x^10 + 1183*x^8 - 7670*x^6 + 14651*x^4 - 3809*x^2 + 13);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 65*x^10 + 1183*x^8 - 7670*x^6 + 14651*x^4 - 3809*x^2 + 13)
\( x^{12} - 65x^{10} + 1183x^{8} - 7670x^{6} + 14651x^{4} - 3809x^{2} + 13 \)
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: |
\(42317611137863236145152\)
\(\medspace = 2^{12}\cdot 7^{8}\cdot 13^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(76.83\) | 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: | $2\cdot 7^{2/3}13^{11/12}\approx 76.83222826035271$ | ||
| Ramified primes: |
\(2\), \(7\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(364=2^{2}\cdot 7\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{364}(1,·)$, $\chi_{364}(99,·)$, $\chi_{364}(165,·)$, $\chi_{364}(289,·)$, $\chi_{364}(205,·)$, $\chi_{364}(239,·)$, $\chi_{364}(337,·)$, $\chi_{364}(275,·)$, $\chi_{364}(277,·)$, $\chi_{364}(219,·)$, $\chi_{364}(123,·)$, $\chi_{364}(319,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{4}a^{6}+\frac{1}{4}$, $\frac{1}{4}a^{7}+\frac{1}{4}a$, $\frac{1}{544}a^{8}+\frac{33}{544}a^{6}+\frac{61}{136}a^{4}-\frac{155}{544}a^{2}+\frac{161}{544}$, $\frac{1}{544}a^{9}+\frac{33}{544}a^{7}+\frac{61}{136}a^{5}-\frac{155}{544}a^{3}+\frac{161}{544}a$, $\frac{1}{5355136}a^{10}-\frac{1}{25024}a^{8}+\frac{308701}{5355136}a^{6}+\frac{577689}{5355136}a^{4}+\frac{1306871}{2677568}a^{2}+\frac{1591689}{5355136}$, $\frac{1}{5355136}a^{11}-\frac{1}{25024}a^{9}+\frac{308701}{5355136}a^{7}+\frac{577689}{5355136}a^{5}+\frac{1306871}{2677568}a^{3}+\frac{1591689}{5355136}a$
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
$C_{3}$, which has order $3$ (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: |
$\frac{59}{232832}a^{10}-\frac{1}{64}a^{8}+\frac{58027}{232832}a^{6}-\frac{277901}{232832}a^{4}+\frac{39923}{116416}a^{2}+\frac{70223}{232832}$, $\frac{95961}{5355136}a^{11}-\frac{59}{232832}a^{10}-\frac{29123}{25024}a^{9}+\frac{1}{64}a^{8}+\frac{6658761}{315008}a^{7}-\frac{58027}{232832}a^{6}-\frac{42975759}{315008}a^{5}+\frac{277901}{232832}a^{4}+\frac{687535513}{2677568}a^{3}-\frac{39923}{116416}a^{2}-\frac{318962267}{5355136}a-\frac{768719}{232832}$, $\frac{28217}{5355136}a^{11}-\frac{59}{232832}a^{10}-\frac{8575}{25024}a^{9}+\frac{1}{64}a^{8}+\frac{33435281}{5355136}a^{7}-\frac{58027}{232832}a^{6}-\frac{217148351}{5355136}a^{5}+\frac{277901}{232832}a^{4}+\frac{207585973}{2677568}a^{3}-\frac{39923}{116416}a^{2}-\frac{96570019}{5355136}a-\frac{70223}{232832}$, $\frac{2685}{2677568}a^{10}-\frac{411}{6256}a^{8}+\frac{3270515}{2677568}a^{6}-\frac{22429059}{2677568}a^{4}+\frac{6331711}{334696}a^{2}-\frac{13844625}{2677568}$, $\frac{6167}{5355136}a^{10}-\frac{1889}{25024}a^{8}+\frac{438831}{315008}a^{6}-\frac{2840545}{315008}a^{4}+\frac{41501003}{2677568}a^{2}-\frac{1229037}{5355136}$, $\frac{3275}{1338784}a^{10}-\frac{499}{3128}a^{8}+\frac{3910277}{1338784}a^{6}-\frac{25463157}{1338784}a^{4}+\frac{89704}{2461}a^{2}-\frac{13363671}{1338784}$, $\frac{9}{157504}a^{10}-\frac{61}{12512}a^{8}+\frac{354125}{2677568}a^{6}-\frac{3202159}{2677568}a^{4}+\frac{2962979}{1338784}a^{2}-\frac{1646247}{2677568}$, $\frac{10885}{1338784}a^{11}-\frac{433}{2677568}a^{10}-\frac{389}{736}a^{9}+\frac{67}{6256}a^{8}+\frac{805231}{83674}a^{7}-\frac{561887}{2677568}a^{6}-\frac{83689399}{1338784}a^{5}+\frac{4682447}{2677568}a^{4}+\frac{161049513}{1338784}a^{3}-\frac{1999609}{334696}a^{2}-\frac{10699501}{334696}a+\frac{6036085}{2677568}$, $\frac{4013}{5355136}a^{11}-\frac{99}{334696}a^{10}-\frac{1253}{25024}a^{9}+\frac{263}{12512}a^{8}+\frac{5206409}{5355136}a^{7}-\frac{610671}{1338784}a^{6}-\frac{38466395}{5355136}a^{5}+\frac{637383}{167348}a^{4}+\frac{49735459}{2677568}a^{3}-\frac{14147663}{1338784}a^{2}-\frac{2983819}{315008}a+\frac{851521}{1338784}$, $\frac{395}{78752}a^{11}+\frac{9}{157504}a^{10}-\frac{1023}{3128}a^{9}-\frac{61}{12512}a^{8}+\frac{8025165}{1338784}a^{7}+\frac{354125}{2677568}a^{6}-\frac{52689157}{1338784}a^{5}-\frac{3202159}{2677568}a^{4}+\frac{6458367}{83674}a^{3}+\frac{2962979}{1338784}a^{2}-\frac{29901423}{1338784}a-\frac{4323815}{2677568}$, $\frac{109391}{2677568}a^{11}+\frac{313}{78752}a^{10}-\frac{33215}{12512}a^{9}-\frac{809}{3128}a^{8}+\frac{129249267}{2677568}a^{7}+\frac{6313575}{1338784}a^{6}-\frac{835966217}{2677568}a^{5}-\frac{40896247}{1338784}a^{4}+\frac{790869385}{1338784}a^{3}+\frac{2367016}{41837}a^{2}-\frac{376087545}{2677568}a-\frac{11086093}{1338784}$
| 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: | \( 16103224.1056 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 16103224.1056 \cdot 3}{2\cdot\sqrt{42317611137863236145152}}\cr\approx \mathstrut & 0.480953923849 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 65*x^10 + 1183*x^8 - 7670*x^6 + 14651*x^4 - 3809*x^2 + 13)
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 - 65*x^10 + 1183*x^8 - 7670*x^6 + 14651*x^4 - 3809*x^2 + 13, 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 - 65*x^10 + 1183*x^8 - 7670*x^6 + 14651*x^4 - 3809*x^2 + 13);
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 - 65*x^10 + 1183*x^8 - 7670*x^6 + 14651*x^4 - 3809*x^2 + 13);
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
| \(\Q(\sqrt{13}) \), 3.3.8281.1, 4.4.35152.1, 6.6.891474493.1 |
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 | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }$ | R | ${\href{/padicField/11.12.0.1}{12} }$ | R | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.1.0.1}{1} }^{12}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ |
| 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
| 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
|
\(7\)
| 7.12.8.2 | $x^{12} - 70 x^{9} + 1519 x^{6} - 4802 x^{3} + 21609$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
|
\(13\)
| 13.12.11.4 | $x^{12} + 13$ | $12$ | $1$ | $11$ | $C_{12}$ | $[\ ]_{12}$ |