Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 42*x^10 + 244*x^9 + 358*x^8 - 4976*x^7 + 3556*x^6 + 61336*x^5 - 169959*x^4 + 17108*x^3 + 437158*x^2 - 492972*x + 440928)
gp: K = bnfinit(y^12 - 4*y^11 - 42*y^10 + 244*y^9 + 358*y^8 - 4976*y^7 + 3556*y^6 + 61336*y^5 - 169959*y^4 + 17108*y^3 + 437158*y^2 - 492972*y + 440928, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 42*x^10 + 244*x^9 + 358*x^8 - 4976*x^7 + 3556*x^6 + 61336*x^5 - 169959*x^4 + 17108*x^3 + 437158*x^2 - 492972*x + 440928);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 - 42*x^10 + 244*x^9 + 358*x^8 - 4976*x^7 + 3556*x^6 + 61336*x^5 - 169959*x^4 + 17108*x^3 + 437158*x^2 - 492972*x + 440928)
\( x^{12} - 4 x^{11} - 42 x^{10} + 244 x^{9} + 358 x^{8} - 4976 x^{7} + 3556 x^{6} + 61336 x^{5} + \cdots + 440928 \)
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: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-426534379192149555604210688\)
\(\medspace = -\,2^{10}\cdot 17^{9}\cdot 37^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(165.64\) | 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: | $2\cdot 17^{3/4}37^{2/3}\approx 185.92359263529752$ | ||
| Ramified primes: |
\(2\), \(17\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{24}a^{8}-\frac{1}{12}a^{7}-\frac{1}{12}a^{6}+\frac{1}{6}a^{5}-\frac{5}{24}a^{4}-\frac{1}{12}a^{3}-\frac{1}{12}a^{2}-\frac{1}{3}a$, $\frac{1}{48}a^{9}-\frac{1}{48}a^{8}+\frac{1}{24}a^{7}+\frac{1}{24}a^{6}-\frac{1}{48}a^{5}+\frac{5}{48}a^{4}-\frac{11}{24}a^{3}+\frac{1}{24}a^{2}-\frac{5}{12}a$, $\frac{1}{902059728}a^{10}+\frac{709935}{150343288}a^{9}+\frac{7140751}{902059728}a^{8}-\frac{5176577}{225514932}a^{7}-\frac{46148867}{902059728}a^{6}+\frac{94819085}{451029864}a^{5}+\frac{91206337}{902059728}a^{4}+\frac{94990837}{225514932}a^{3}+\frac{55497707}{150343288}a^{2}+\frac{103929151}{225514932}a+\frac{3628757}{18792911}$, $\frac{1}{810886747171584}a^{11}-\frac{35713}{810886747171584}a^{10}+\frac{7218027470803}{810886747171584}a^{9}+\frac{14250007609901}{810886747171584}a^{8}-\frac{3197117162881}{42678249851136}a^{7}-\frac{27500557257769}{810886747171584}a^{6}+\frac{68452214460905}{810886747171584}a^{5}-\frac{139961637359725}{810886747171584}a^{4}+\frac{63278684917673}{405443373585792}a^{3}-\frac{59371177615049}{135147791195264}a^{2}+\frac{57968860642273}{202721686792896}a-\frac{2956807523251}{8446736949704}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
$C_{2}\times C_{78}$, which has order $156$ (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: | $6$ | 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: |
$\frac{592}{56378733}a^{11}-\frac{1227}{37585822}a^{10}-\frac{27905}{56378733}a^{9}+\frac{39205}{18792911}a^{8}+\frac{127984}{18792911}a^{7}-\frac{1750421}{37585822}a^{6}-\frac{551123}{18792911}a^{5}+\frac{12624920}{18792911}a^{4}-\frac{48773812}{56378733}a^{3}-\frac{32021299}{18792911}a^{2}+\frac{100417142}{56378733}a-\frac{46047589}{18792911}$, $\frac{907395955}{810886747171584}a^{11}-\frac{1018103507}{810886747171584}a^{10}-\frac{32195921335}{810886747171584}a^{9}+\frac{23803354525}{270295582390528}a^{8}+\frac{3113410095}{14226083283712}a^{7}-\frac{277931693209}{270295582390528}a^{6}+\frac{494543698617}{270295582390528}a^{5}-\frac{113249637725}{270295582390528}a^{4}-\frac{22550591268245}{405443373585792}a^{3}+\frac{266522816170111}{405443373585792}a^{2}-\frac{110089573251917}{202721686792896}a-\frac{28880828648777}{8446736949704}$, $\frac{794763145}{270295582390528}a^{11}-\frac{37282945787}{810886747171584}a^{10}-\frac{69168353215}{810886747171584}a^{9}+\frac{1914995356895}{810886747171584}a^{8}-\frac{106208184475}{42678249851136}a^{7}-\frac{33444125848627}{810886747171584}a^{6}+\frac{86727788307347}{810886747171584}a^{5}+\frac{326848811870945}{810886747171584}a^{4}-\frac{845724473262125}{405443373585792}a^{3}+\frac{285118036899911}{405443373585792}a^{2}+\frac{341871670381833}{67573895597632}a-\frac{59501198198225}{8446736949704}$, $\frac{35\!\cdots\!79}{405443373585792}a^{11}-\frac{31\!\cdots\!29}{135147791195264}a^{10}-\frac{16\!\cdots\!47}{405443373585792}a^{9}+\frac{21\!\cdots\!61}{135147791195264}a^{8}+\frac{43\!\cdots\!15}{7113041641856}a^{7}-\frac{48\!\cdots\!13}{135147791195264}a^{6}-\frac{43\!\cdots\!75}{135147791195264}a^{5}+\frac{70\!\cdots\!23}{135147791195264}a^{4}-\frac{12\!\cdots\!01}{202721686792896}a^{3}-\frac{87\!\cdots\!15}{67573895597632}a^{2}+\frac{13\!\cdots\!59}{101360843396448}a-\frac{80\!\cdots\!13}{4223368474852}$, $\frac{30\!\cdots\!75}{810886747171584}a^{11}-\frac{30\!\cdots\!11}{810886747171584}a^{10}-\frac{16\!\cdots\!75}{810886747171584}a^{9}+\frac{22\!\cdots\!99}{810886747171584}a^{8}+\frac{16\!\cdots\!29}{42678249851136}a^{7}-\frac{48\!\cdots\!79}{810886747171584}a^{6}-\frac{31\!\cdots\!41}{810886747171584}a^{5}+\frac{83\!\cdots\!53}{810886747171584}a^{4}+\frac{12\!\cdots\!93}{135147791195264}a^{3}-\frac{91\!\cdots\!93}{405443373585792}a^{2}+\frac{56\!\cdots\!39}{202721686792896}a-\frac{17\!\cdots\!65}{8446736949704}$, $\frac{16\!\cdots\!93}{67573895597632}a^{11}-\frac{45\!\cdots\!23}{202721686792896}a^{10}+\frac{66\!\cdots\!09}{202721686792896}a^{9}+\frac{22\!\cdots\!65}{67573895597632}a^{8}-\frac{38\!\cdots\!53}{3556520820928}a^{7}-\frac{29\!\cdots\!13}{67573895597632}a^{6}+\frac{22\!\cdots\!25}{67573895597632}a^{5}-\frac{41\!\cdots\!97}{67573895597632}a^{4}-\frac{51\!\cdots\!83}{33786947798816}a^{3}+\frac{22\!\cdots\!11}{101360843396448}a^{2}-\frac{11\!\cdots\!89}{50680421698224}a+\frac{40\!\cdots\!37}{2111684237426}$
| 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: | \( 301266893.48755616 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 301266893.48755616 \cdot 156}{2\cdot\sqrt{426534379192149555604210688}}\cr\approx \mathstrut & 44.5685314619468 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 42*x^10 + 244*x^9 + 358*x^8 - 4976*x^7 + 3556*x^6 + 61336*x^5 - 169959*x^4 + 17108*x^3 + 437158*x^2 - 492972*x + 440928)
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 - 4*x^11 - 42*x^10 + 244*x^9 + 358*x^8 - 4976*x^7 + 3556*x^6 + 61336*x^5 - 169959*x^4 + 17108*x^3 + 437158*x^2 - 492972*x + 440928, 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 - 4*x^11 - 42*x^10 + 244*x^9 + 358*x^8 - 4976*x^7 + 3556*x^6 + 61336*x^5 - 169959*x^4 + 17108*x^3 + 437158*x^2 - 492972*x + 440928);
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 - 4*x^11 - 42*x^10 + 244*x^9 + 358*x^8 - 4976*x^7 + 3556*x^6 + 61336*x^5 - 169959*x^4 + 17108*x^3 + 437158*x^2 - 492972*x + 440928);
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 24 |
| The 9 conjugacy class representatives for $D_{12}$ |
| Character table for $D_{12}$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 3.1.5476.1, 4.2.19652.1, 6.2.147324047888.4 |
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
| Galois closure: | deg 24 |
| Degree 12 sibling: | deg 12 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.2.0.1}{2} }^{6}$ | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.2.0.1}{2} }^{6}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{5}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 | $[\ ]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(37\)
| 37.12.8.2 | $x^{12} + 8214 x^{6} - 1215672 x^{3} + 3748322$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.68.2t1.a.a | $1$ | $ 2^{2} \cdot 17 $ | \(\Q(\sqrt{-17}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 2.5476.3t2.a.a | $2$ | $ 2^{2} \cdot 37^{2}$ | 3.1.5476.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.1582564.6t3.a.a | $2$ | $ 2^{2} \cdot 17^{2} \cdot 37^{2}$ | 6.2.147324047888.4 | $D_{6}$ (as 6T3) | $1$ | $0$ |
| * | 2.1156.4t3.a.a | $2$ | $ 2^{2} \cdot 17^{2}$ | 4.2.19652.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
| * | 2.1582564.12t12.a.b | $2$ | $ 2^{2} \cdot 17^{2} \cdot 37^{2}$ | 12.2.426534379192149555604210688.1 | $D_{12}$ (as 12T12) | $1$ | $0$ |
| * | 2.1582564.12t12.a.a | $2$ | $ 2^{2} \cdot 17^{2} \cdot 37^{2}$ | 12.2.426534379192149555604210688.1 | $D_{12}$ (as 12T12) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.