Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 30*x^10 - 26*x^9 + 761*x^8 - 336*x^7 - 1632*x^6 + 4656*x^5 + 1424*x^4 - 5824*x^3 + 11904*x^2 + 5120*x + 4096)
gp: K = bnfinit(y^12 - 2*y^11 - 30*y^10 - 26*y^9 + 761*y^8 - 336*y^7 - 1632*y^6 + 4656*y^5 + 1424*y^4 - 5824*y^3 + 11904*y^2 + 5120*y + 4096, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 - 30*x^10 - 26*x^9 + 761*x^8 - 336*x^7 - 1632*x^6 + 4656*x^5 + 1424*x^4 - 5824*x^3 + 11904*x^2 + 5120*x + 4096);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 2*x^11 - 30*x^10 - 26*x^9 + 761*x^8 - 336*x^7 - 1632*x^6 + 4656*x^5 + 1424*x^4 - 5824*x^3 + 11904*x^2 + 5120*x + 4096)
\( x^{12} - 2 x^{11} - 30 x^{10} - 26 x^{9} + 761 x^{8} - 336 x^{7} - 1632 x^{6} + 4656 x^{5} + 1424 x^{4} + \cdots + 4096 \)
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: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(396229730290715706724352\)
\(\medspace = 2^{12}\cdot 13^{8}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(92.58\) | 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: | $2\cdot 13^{2/3}17^{3/4}\approx 92.57539808355045$ | ||
| Ramified primes: |
\(2\), \(13\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{7}-\frac{1}{8}a^{3}$, $\frac{1}{96}a^{8}-\frac{1}{16}a^{7}+\frac{1}{16}a^{6}+\frac{1}{16}a^{5}+\frac{3}{32}a^{4}-\frac{1}{8}a^{3}-\frac{3}{8}a^{2}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{192}a^{9}-\frac{1}{32}a^{7}-\frac{1}{32}a^{6}-\frac{1}{64}a^{5}-\frac{1}{32}a^{4}+\frac{1}{16}a^{3}-\frac{3}{8}a^{2}-\frac{1}{3}a$, $\frac{1}{3576576}a^{10}+\frac{547}{223536}a^{9}-\frac{917}{596096}a^{8}-\frac{17145}{596096}a^{7}+\frac{136847}{1192192}a^{6}-\frac{2465}{596096}a^{5}-\frac{6689}{298048}a^{4}+\frac{34829}{149024}a^{3}-\frac{19333}{55884}a^{2}-\frac{16795}{55884}a+\frac{2287}{4657}$, $\frac{1}{32196337152}a^{11}-\frac{725}{5366056192}a^{10}-\frac{33176587}{16098168576}a^{9}+\frac{19487759}{16098168576}a^{8}-\frac{498909797}{10732112384}a^{7}-\frac{334072569}{2683028096}a^{6}-\frac{38392217}{670757024}a^{5}+\frac{406583}{83844628}a^{4}+\frac{488349665}{2012271072}a^{3}+\frac{31477967}{167689256}a^{2}-\frac{91275029}{251533884}a+\frac{26771389}{62883471}$
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
$C_{3}\times C_{24}$, which has order $72$ (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: | $5$ | 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: |
$\frac{224073}{10732112384}a^{11}+\frac{327077}{8049084288}a^{10}-\frac{5188059}{5366056192}a^{9}-\frac{43133005}{16098168576}a^{8}+\frac{194616861}{10732112384}a^{7}+\frac{336614251}{5366056192}a^{6}-\frac{452706271}{2683028096}a^{5}+\frac{49355293}{1341514048}a^{4}+\frac{4203231}{83844628}a^{3}-\frac{263993201}{503067768}a^{2}-\frac{5258065}{20961157}a-\frac{16815671}{62883471}$, $\frac{7884643}{8049084288}a^{11}-\frac{35105143}{16098168576}a^{10}-\frac{29776337}{1006135536}a^{9}-\frac{133737509}{8049084288}a^{8}+\frac{64519839}{83844628}a^{7}-\frac{2701356797}{5366056192}a^{6}-\frac{5559721415}{2683028096}a^{5}+\frac{7583662759}{1341514048}a^{4}+\frac{2769845237}{2012271072}a^{3}-\frac{417829415}{62883471}a^{2}+\frac{3443012183}{251533884}a+\frac{627232027}{62883471}$, $\frac{366210735461}{2683028096}a^{11}-\frac{8607868024583}{16098168576}a^{10}-\frac{1041941049173}{335378512}a^{9}+\frac{20975265125659}{8049084288}a^{8}+\frac{66720150022845}{670757024}a^{7}-\frac{12\!\cdots\!41}{5366056192}a^{6}+\frac{561685748895185}{2683028096}a^{5}+\frac{450350399691251}{1341514048}a^{4}-\frac{439371961251029}{670757024}a^{3}+\frac{42023266376453}{62883471}a^{2}+\frac{18622191697993}{83844628}a+\frac{16687769765071}{62883471}$, $\frac{301757967722165}{8049084288}a^{11}+\frac{32521656622843}{16098168576}a^{10}-\frac{11\!\cdots\!05}{1006135536}a^{9}-\frac{92\!\cdots\!53}{2683028096}a^{8}+\frac{15\!\cdots\!73}{670757024}a^{7}+\frac{22\!\cdots\!93}{5366056192}a^{6}-\frac{16\!\cdots\!45}{2683028096}a^{5}+\frac{10\!\cdots\!49}{1341514048}a^{4}+\frac{30\!\cdots\!31}{2012271072}a^{3}-\frac{38\!\cdots\!14}{62883471}a^{2}-\frac{77\!\cdots\!33}{251533884}a-\frac{783713988260759}{20961157}$, $\frac{12\!\cdots\!23}{2299738368}a^{11}-\frac{48\!\cdots\!45}{574934592}a^{10}-\frac{67\!\cdots\!43}{383289728}a^{9}-\frac{81\!\cdots\!55}{383289728}a^{8}+\frac{33\!\cdots\!17}{766579456}a^{7}+\frac{32\!\cdots\!31}{383289728}a^{6}-\frac{31\!\cdots\!99}{191644864}a^{5}+\frac{90\!\cdots\!73}{95822432}a^{4}+\frac{25\!\cdots\!95}{71866824}a^{3}-\frac{23\!\cdots\!71}{17966706}a^{2}-\frac{18\!\cdots\!86}{2994451}a-\frac{25\!\cdots\!35}{2994451}$
| 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: | \( 19501292.62022837 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 19501292.62022837 \cdot 72}{2\cdot\sqrt{396229730290715706724352}}\cr\approx \mathstrut & 68.6232971039376 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 30*x^10 - 26*x^9 + 761*x^8 - 336*x^7 - 1632*x^6 + 4656*x^5 + 1424*x^4 - 5824*x^3 + 11904*x^2 + 5120*x + 4096)
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 - 30*x^10 - 26*x^9 + 761*x^8 - 336*x^7 - 1632*x^6 + 4656*x^5 + 1424*x^4 - 5824*x^3 + 11904*x^2 + 5120*x + 4096, 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 - 30*x^10 - 26*x^9 + 761*x^8 - 336*x^7 - 1632*x^6 + 4656*x^5 + 1424*x^4 - 5824*x^3 + 11904*x^2 + 5120*x + 4096);
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 - 30*x^10 - 26*x^9 + 761*x^8 - 336*x^7 - 1632*x^6 + 4656*x^5 + 1424*x^4 - 5824*x^3 + 11904*x^2 + 5120*x + 4096);
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
| \(\Q(\sqrt{-17}) \), 3.1.676.1, 4.0.78608.2, 6.0.8980492352.3 |
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: | 12.2.99057432572678926681088.1 |
| Minimal sibling: | 12.2.99057432572678926681088.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.2.0.1}{2} }^{5}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.2.0.1}{2} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{5}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.2.0.1}{2} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 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 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
|
\(13\)
| 13.6.4.2 | $x^{6} - 156 x^{3} + 338$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 13.6.4.2 | $x^{6} - 156 x^{3} + 338$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
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) | $1$ | $-1$ | |
| 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 1.68.2t1.a.a | $1$ | $ 2^{2} \cdot 17 $ | \(\Q(\sqrt{-17}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 2.676.3t2.b.a | $2$ | $ 2^{2} \cdot 13^{2}$ | 3.1.676.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.195364.6t3.b.a | $2$ | $ 2^{2} \cdot 13^{2} \cdot 17^{2}$ | 6.0.8980492352.3 | $D_{6}$ (as 6T3) | $1$ | $0$ |
| * | 2.1156.4t3.c.a | $2$ | $ 2^{2} \cdot 17^{2}$ | 4.0.78608.2 | $D_{4}$ (as 4T3) | $1$ | $0$ |
| * | 2.195364.12t12.a.b | $2$ | $ 2^{2} \cdot 13^{2} \cdot 17^{2}$ | 12.0.396229730290715706724352.2 | $D_{12}$ (as 12T12) | $1$ | $0$ |
| * | 2.195364.12t12.a.a | $2$ | $ 2^{2} \cdot 13^{2} \cdot 17^{2}$ | 12.0.396229730290715706724352.2 | $D_{12}$ (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 *.