Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 18*x^10 + 78*x^9 - 127*x^8 + 86*x^7 + 158*x^6 - 450*x^5 + 168*x^4 + 184*x^3 - 50*x^2 - 16*x + 4)
gp: K = bnfinit(y^12 - 2*y^11 - 18*y^10 + 78*y^9 - 127*y^8 + 86*y^7 + 158*y^6 - 450*y^5 + 168*y^4 + 184*y^3 - 50*y^2 - 16*y + 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 - 18*x^10 + 78*x^9 - 127*x^8 + 86*x^7 + 158*x^6 - 450*x^5 + 168*x^4 + 184*x^3 - 50*x^2 - 16*x + 4);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 2*x^11 - 18*x^10 + 78*x^9 - 127*x^8 + 86*x^7 + 158*x^6 - 450*x^5 + 168*x^4 + 184*x^3 - 50*x^2 - 16*x + 4)
\( x^{12} - 2 x^{11} - 18 x^{10} + 78 x^{9} - 127 x^{8} + 86 x^{7} + 158 x^{6} - 450 x^{5} + 168 x^{4} + \cdots + 4 \)
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: | $[8, 2]$ | 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: | $2^{13/6}17^{11/12}\approx 60.275825724785875$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}$, $\frac{1}{8}a^{10}+\frac{1}{8}a^{8}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{26312672}a^{11}-\frac{320167}{26312672}a^{10}+\frac{5253761}{26312672}a^{9}-\frac{6520215}{26312672}a^{8}+\frac{622389}{6578168}a^{7}-\frac{2618215}{13156336}a^{6}-\frac{860427}{6578168}a^{5}-\frac{3818323}{13156336}a^{4}-\frac{6514077}{13156336}a^{3}-\frac{2388931}{13156336}a^{2}-\frac{2328053}{6578168}a+\frac{2028997}{6578168}$
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 $2$
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: | $9$ | 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: |
$\frac{91733311}{26312672}a^{11}-\frac{159319365}{26312672}a^{10}-\frac{1692393409}{26312672}a^{9}+\frac{6708185611}{26312672}a^{8}-\frac{2474408321}{6578168}a^{7}+\frac{2670956423}{13156336}a^{6}+\frac{3950564217}{6578168}a^{5}-\frac{18523189185}{13156336}a^{4}+\frac{2883753877}{13156336}a^{3}+\frac{9034075431}{13156336}a^{2}+\frac{76104989}{6578168}a-\frac{325358545}{6578168}$, $\frac{24618093}{26312672}a^{11}-\frac{48623703}{26312672}a^{10}-\frac{441493171}{26312672}a^{9}+\frac{1903912921}{26312672}a^{8}-\frac{782588531}{6578168}a^{7}+\frac{1126630421}{13156336}a^{6}+\frac{900285255}{6578168}a^{5}-\frac{5371602155}{13156336}a^{4}+\frac{2142773759}{13156336}a^{3}+\frac{1721596109}{13156336}a^{2}-\frac{179955145}{6578168}a-\frac{22146107}{6578168}$, $\frac{49252317}{26312672}a^{11}-\frac{107866655}{26312672}a^{10}-\frac{861371491}{26312672}a^{9}+\frac{3996043617}{26312672}a^{8}-\frac{1774205847}{6578168}a^{7}+\frac{2974271717}{13156336}a^{6}+\frac{1506584035}{6578168}a^{5}-\frac{11406291523}{13156336}a^{4}+\frac{6608720367}{13156336}a^{3}+\frac{2254895077}{13156336}a^{2}-\frac{569473497}{6578168}a+\frac{28143357}{6578168}$, $\frac{4450751}{26312672}a^{11}-\frac{4397837}{26312672}a^{10}-\frac{87045393}{26312672}a^{9}+\frac{264243059}{26312672}a^{8}-\frac{63464097}{6578168}a^{7}-\frac{35231513}{13156336}a^{6}+\frac{241699877}{6578168}a^{5}-\frac{645271513}{13156336}a^{4}-\frac{436753899}{13156336}a^{3}+\frac{509101455}{13156336}a^{2}+\frac{76310909}{6578168}a-\frac{11096249}{6578168}$, $\frac{69088337}{26312672}a^{11}-\frac{166415251}{26312672}a^{10}-\frac{1174448815}{26312672}a^{9}+\frac{5866517885}{26312672}a^{8}-\frac{2797972887}{6578168}a^{7}+\frac{5303107289}{13156336}a^{6}+\frac{1606353011}{6578168}a^{5}-\frac{16796139383}{13156336}a^{4}+\frac{12740026747}{13156336}a^{3}+\frac{895814561}{13156336}a^{2}-\frac{984684693}{6578168}a+\frac{159769753}{6578168}$, $\frac{16883285}{13156336}a^{11}-\frac{28897879}{13156336}a^{10}-\frac{312377659}{13156336}a^{9}+\frac{1227135617}{13156336}a^{8}-\frac{446944385}{3289084}a^{7}+\frac{462434405}{6578168}a^{6}+\frac{738946867}{3289084}a^{5}-\frac{3383999867}{6578168}a^{4}+\frac{438189199}{6578168}a^{3}+\frac{1708435885}{6578168}a^{2}+\frac{26033611}{3289084}a-\frac{56778599}{3289084}$, $\frac{36076711}{26312672}a^{11}-\frac{62522973}{26312672}a^{10}-\frac{665995401}{26312672}a^{9}+\frac{2635897091}{26312672}a^{8}-\frac{969816033}{6578168}a^{7}+\frac{1038744815}{13156336}a^{6}+\frac{1559365225}{6578168}a^{5}-\frac{7286083225}{13156336}a^{4}+\frac{1111033933}{13156336}a^{3}+\frac{3538318543}{13156336}a^{2}+\frac{47349845}{6578168}a-\frac{121296105}{6578168}$, $\frac{5653091}{26312672}a^{11}-\frac{12753593}{26312672}a^{10}-\frac{97430013}{26312672}a^{9}+\frac{464842663}{26312672}a^{8}-\frac{214096129}{6578168}a^{7}+\frac{383444539}{13156336}a^{6}+\frac{152553201}{6578168}a^{5}-\frac{1322116933}{13156336}a^{4}+\frac{881392689}{13156336}a^{3}+\frac{151013699}{13156336}a^{2}-\frac{63918119}{6578168}a+\frac{6397595}{6578168}$, $\frac{6629367}{26312672}a^{11}-\frac{9302829}{26312672}a^{10}-\frac{132121609}{26312672}a^{9}+\frac{450986387}{26312672}a^{8}-\frac{110780237}{6578168}a^{7}-\frac{111911137}{13156336}a^{6}+\frac{443370921}{6578168}a^{5}-\frac{1283776329}{13156336}a^{4}-\frac{698396531}{13156336}a^{3}+\frac{1611939599}{13156336}a^{2}+\frac{127403805}{6578168}a-\frac{75460585}{6578168}$
| 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: | \( 678751.076549 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{2}\cdot 678751.076549 \cdot 2}{2\cdot\sqrt{4492085992834072576}}\cr\approx \mathstrut & 3.23657893121 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 18*x^10 + 78*x^9 - 127*x^8 + 86*x^7 + 158*x^6 - 450*x^5 + 168*x^4 + 184*x^3 - 50*x^2 - 16*x + 4)
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 - 18*x^10 + 78*x^9 - 127*x^8 + 86*x^7 + 158*x^6 - 450*x^5 + 168*x^4 + 184*x^3 - 50*x^2 - 16*x + 4, 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 - 18*x^10 + 78*x^9 - 127*x^8 + 86*x^7 + 158*x^6 - 450*x^5 + 168*x^4 + 184*x^3 - 50*x^2 - 16*x + 4);
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 - 18*x^10 + 78*x^9 - 127*x^8 + 86*x^7 + 158*x^6 - 450*x^5 + 168*x^4 + 184*x^3 - 50*x^2 - 16*x + 4);
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
$D_6\wr C_2$ (as 12T125):
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 288 |
| The 27 conjugacy class representatives for $D_6\wr C_2$ |
| Character table for $D_6\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.157216.1, 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 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.2.561510749104259072.4 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{5}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.6.8.3 | $x^{6} + 2 x^{3} + 6$ | $6$ | $1$ | $8$ | $D_{6}$ | $[2]_{3}^{2}$ | |
|
\(17\)
| 17.12.11.2 | $x^{12} + 34$ | $12$ | $1$ | $11$ | $S_3 \times C_4$ | $[\ ]_{12}^{2}$ |