Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 17*x^10 + 119*x^8 - 255*x^6 - 170*x^4 + 612*x^2 - 34)
gp: K = bnfinit(y^12 - 17*y^10 + 119*y^8 - 255*y^6 - 170*y^4 + 612*y^2 - 34, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 17*x^10 + 119*x^8 - 255*x^6 - 170*x^4 + 612*x^2 - 34);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 17*x^10 + 119*x^8 - 255*x^6 - 170*x^4 + 612*x^2 - 34)
\( x^{12} - 17x^{10} + 119x^{8} - 255x^{6} - 170x^{4} + 612x^{2} - 34 \)
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: | $[6, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-4492085992834072576\)
\(\medspace = -\,2^{17}\cdot 17^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(35.84\) | 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: | $2^{31/12}17^{11/12}\approx 80.45857442045158$ | ||
| Ramified primes: |
\(2\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-34}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}$, $\frac{1}{4}a^{8}+\frac{1}{4}a^{6}-\frac{1}{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}+\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{884192}a^{10}-\frac{6379}{55262}a^{8}+\frac{427559}{884192}a^{6}+\frac{28113}{110524}a^{4}+\frac{96543}{442096}a^{2}+\frac{186145}{442096}$, $\frac{1}{1768384}a^{11}-\frac{1}{1768384}a^{10}-\frac{6379}{110524}a^{9}+\frac{6379}{110524}a^{8}+\frac{427559}{1768384}a^{7}-\frac{427559}{1768384}a^{6}-\frac{82411}{221048}a^{5}+\frac{82411}{221048}a^{4}+\frac{96543}{884192}a^{3}-\frac{96543}{884192}a^{2}-\frac{255951}{884192}a+\frac{255951}{884192}$
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
$C_{2}$, which has order $2$ (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: | $8$ | 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: |
$\frac{9505}{884192}a^{11}+\frac{1889}{884192}a^{10}-\frac{9981}{55262}a^{9}-\frac{2815}{55262}a^{8}+\frac{1086055}{884192}a^{7}+\frac{391655}{884192}a^{6}-\frac{254015}{110524}a^{5}-\frac{167111}{110524}a^{4}-\frac{1476369}{442096}a^{3}-\frac{215921}{442096}a^{2}+\frac{2692609}{442096}a+\frac{603681}{442096}$, $\frac{4589}{884192}a^{11}+\frac{1695}{884192}a^{10}-\frac{20377}{221048}a^{9}-\frac{7105}{221048}a^{8}+\frac{598823}{884192}a^{7}+\frac{227685}{884192}a^{6}-\frac{191999}{110524}a^{5}-\frac{94833}{110524}a^{4}+\frac{55635}{442096}a^{3}+\frac{506961}{442096}a^{2}+\frac{1305697}{442096}a+\frac{190803}{442096}$, $\frac{7609}{221048}a^{10}-\frac{59307}{110524}a^{8}+\frac{740897}{221048}a^{6}-\frac{117909}{27631}a^{4}-\frac{1162581}{110524}a^{2}+\frac{509603}{110524}$, $\frac{9505}{884192}a^{11}-\frac{1889}{884192}a^{10}-\frac{9981}{55262}a^{9}+\frac{2815}{55262}a^{8}+\frac{1086055}{884192}a^{7}-\frac{391655}{884192}a^{6}-\frac{254015}{110524}a^{5}+\frac{167111}{110524}a^{4}-\frac{1476369}{442096}a^{3}+\frac{215921}{442096}a^{2}+\frac{2692609}{442096}a-\frac{603681}{442096}$, $\frac{476}{27631}a^{10}-\frac{7166}{27631}a^{8}+\frac{43400}{27631}a^{6}-\frac{43452}{27631}a^{4}-\frac{157556}{27631}a^{2}+\frac{233485}{27631}$, $\frac{7489}{442096}a^{11}+\frac{4933}{221048}a^{10}-\frac{69549}{221048}a^{9}-\frac{72523}{221048}a^{8}+\frac{1048525}{442096}a^{7}+\frac{54190}{27631}a^{6}-\frac{341535}{55262}a^{5}-\frac{53822}{27631}a^{4}-\frac{700625}{221048}a^{3}-\frac{443393}{110524}a^{2}+\frac{3371675}{221048}a+\frac{122405}{27631}$, $\frac{7489}{442096}a^{11}-\frac{4933}{221048}a^{10}-\frac{69549}{221048}a^{9}+\frac{72523}{221048}a^{8}+\frac{1048525}{442096}a^{7}-\frac{54190}{27631}a^{6}-\frac{341535}{55262}a^{5}+\frac{53822}{27631}a^{4}-\frac{700625}{221048}a^{3}+\frac{443393}{110524}a^{2}+\frac{3371675}{221048}a-\frac{122405}{27631}$, $\frac{28315}{442096}a^{11}+\frac{6713}{221048}a^{10}-\frac{228983}{221048}a^{9}-\frac{100249}{221048}a^{8}+\frac{3015631}{442096}a^{7}+\frac{293031}{110524}a^{6}-\frac{637397}{55262}a^{5}-\frac{52423}{27631}a^{4}-\frac{3401291}{221048}a^{3}-\frac{1235341}{110524}a^{2}+\frac{4724433}{221048}a-\frac{176081}{55262}$
| 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: | \( 737857.290597 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{3}\cdot 737857.290597 \cdot 2}{2\cdot\sqrt{4492085992834072576}}\cr\approx \mathstrut & 5.52672583652 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 17*x^10 + 119*x^8 - 255*x^6 - 170*x^4 + 612*x^2 - 34)
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 - 17*x^10 + 119*x^8 - 255*x^6 - 170*x^4 + 612*x^2 - 34, 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 - 17*x^10 + 119*x^8 - 255*x^6 - 170*x^4 + 612*x^2 - 34);
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 - 17*x^10 + 119*x^8 - 255*x^6 - 170*x^4 + 612*x^2 - 34);
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
$S_4^2:C_4$ (as 12T238):
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 2304 |
| The 40 conjugacy class representatives for $S_4^2:C_4$ |
| Character table for $S_4^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 6.4.45435424.1 |
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
| Degree 12 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.4.2246042996417036288.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | R | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.6.11.11 | $x^{6} + 4 x^{4} + 4 x + 14$ | $6$ | $1$ | $11$ | $S_4\times C_2$ | $[8/3, 8/3, 3]_{3}^{2}$ | |
|
\(17\)
| 17.12.11.2 | $x^{12} + 34$ | $12$ | $1$ | $11$ | $S_3 \times C_4$ | $[\ ]_{12}^{2}$ |