Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 34*x^10 - 192*x^9 + 2255*x^8 - 8416*x^7 + 20394*x^6 - 2272*x^5 + 143792*x^4 - 271424*x^3 + 129376*x^2 + 327680*x + 1048576)
gp: K = bnfinit(y^12 - 4*y^11 + 34*y^10 - 192*y^9 + 2255*y^8 - 8416*y^7 + 20394*y^6 - 2272*y^5 + 143792*y^4 - 271424*y^3 + 129376*y^2 + 327680*y + 1048576, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 34*x^10 - 192*x^9 + 2255*x^8 - 8416*x^7 + 20394*x^6 - 2272*x^5 + 143792*x^4 - 271424*x^3 + 129376*x^2 + 327680*x + 1048576);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 34*x^10 - 192*x^9 + 2255*x^8 - 8416*x^7 + 20394*x^6 - 2272*x^5 + 143792*x^4 - 271424*x^3 + 129376*x^2 + 327680*x + 1048576)
\( x^{12} - 4 x^{11} + 34 x^{10} - 192 x^{9} + 2255 x^{8} - 8416 x^{7} + 20394 x^{6} - 2272 x^{5} + \cdots + 1048576 \)
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: |
\(1462520021180890181009408\)
\(\medspace = 2^{18}\cdot 17^{9}\cdot 19^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(103.22\) | 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: | $2^{3/2}17^{3/4}19^{1/2}\approx 103.21872376945974$ | ||
| Ramified primes: |
\(2\), \(17\), \(19\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{9}-\frac{1}{16}a^{7}-\frac{1}{32}a^{5}+\frac{1}{8}a^{4}-\frac{5}{16}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{53504}a^{10}+\frac{349}{26752}a^{9}+\frac{1271}{26752}a^{8}-\frac{663}{13376}a^{7}-\frac{1913}{53504}a^{6}-\frac{6359}{26752}a^{5}-\frac{4701}{26752}a^{4}-\frac{4683}{13376}a^{3}-\frac{1135}{6688}a^{2}+\frac{387}{3344}a-\frac{51}{209}$, $\frac{1}{78\!\cdots\!40}a^{11}+\frac{48\!\cdots\!43}{19\!\cdots\!60}a^{10}-\frac{44\!\cdots\!87}{39\!\cdots\!20}a^{9}+\frac{13\!\cdots\!67}{22\!\cdots\!20}a^{8}+\frac{85\!\cdots\!49}{71\!\cdots\!40}a^{7}+\frac{82\!\cdots\!73}{49\!\cdots\!40}a^{6}-\frac{75\!\cdots\!19}{39\!\cdots\!20}a^{5}+\frac{69\!\cdots\!95}{61\!\cdots\!08}a^{4}-\frac{96\!\cdots\!33}{49\!\cdots\!40}a^{3}+\frac{55\!\cdots\!17}{12\!\cdots\!60}a^{2}-\frac{59\!\cdots\!83}{22\!\cdots\!20}a+\frac{33\!\cdots\!79}{19\!\cdots\!15}$
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
$C_{4}\times C_{8}$, which has order $32$ (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: |
$\frac{12\!\cdots\!65}{47\!\cdots\!16}a^{11}-\frac{11\!\cdots\!35}{23\!\cdots\!08}a^{10}+\frac{51\!\cdots\!59}{23\!\cdots\!08}a^{9}-\frac{19\!\cdots\!05}{10\!\cdots\!64}a^{8}+\frac{56\!\cdots\!25}{43\!\cdots\!56}a^{7}-\frac{24\!\cdots\!23}{23\!\cdots\!08}a^{6}+\frac{83\!\cdots\!11}{23\!\cdots\!08}a^{5}-\frac{11\!\cdots\!75}{11\!\cdots\!04}a^{4}+\frac{82\!\cdots\!57}{59\!\cdots\!52}a^{3}-\frac{20\!\cdots\!65}{29\!\cdots\!76}a^{2}+\frac{22\!\cdots\!92}{40\!\cdots\!57}a-\frac{24\!\cdots\!09}{93\!\cdots\!43}$, $\frac{90\!\cdots\!47}{15\!\cdots\!48}a^{11}-\frac{33\!\cdots\!79}{39\!\cdots\!12}a^{10}-\frac{51\!\cdots\!53}{78\!\cdots\!24}a^{9}-\frac{14\!\cdots\!41}{44\!\cdots\!24}a^{8}+\frac{17\!\cdots\!87}{14\!\cdots\!68}a^{7}-\frac{12\!\cdots\!83}{98\!\cdots\!28}a^{6}-\frac{10\!\cdots\!41}{78\!\cdots\!24}a^{5}-\frac{89\!\cdots\!95}{24\!\cdots\!32}a^{4}+\frac{11\!\cdots\!57}{98\!\cdots\!28}a^{3}-\frac{34\!\cdots\!93}{24\!\cdots\!32}a^{2}-\frac{14\!\cdots\!89}{44\!\cdots\!24}a-\frac{20\!\cdots\!79}{38\!\cdots\!63}$, $\frac{23\!\cdots\!61}{78\!\cdots\!40}a^{11}+\frac{12\!\cdots\!03}{19\!\cdots\!60}a^{10}+\frac{76\!\cdots\!13}{39\!\cdots\!20}a^{9}+\frac{67\!\cdots\!63}{10\!\cdots\!80}a^{8}-\frac{41\!\cdots\!11}{71\!\cdots\!40}a^{7}+\frac{21\!\cdots\!33}{49\!\cdots\!40}a^{6}+\frac{20\!\cdots\!61}{39\!\cdots\!20}a^{5}+\frac{16\!\cdots\!13}{12\!\cdots\!16}a^{4}-\frac{20\!\cdots\!53}{49\!\cdots\!40}a^{3}+\frac{53\!\cdots\!57}{12\!\cdots\!60}a^{2}+\frac{24\!\cdots\!97}{22\!\cdots\!20}a+\frac{36\!\cdots\!09}{19\!\cdots\!15}$, $\frac{39\!\cdots\!33}{98\!\cdots\!80}a^{11}-\frac{37\!\cdots\!77}{49\!\cdots\!40}a^{10}+\frac{42\!\cdots\!19}{49\!\cdots\!40}a^{9}-\frac{13\!\cdots\!67}{22\!\cdots\!20}a^{8}+\frac{61\!\cdots\!97}{89\!\cdots\!80}a^{7}-\frac{78\!\cdots\!93}{49\!\cdots\!40}a^{6}-\frac{36\!\cdots\!97}{49\!\cdots\!40}a^{5}+\frac{64\!\cdots\!55}{49\!\cdots\!64}a^{4}+\frac{79\!\cdots\!67}{12\!\cdots\!60}a^{3}+\frac{49\!\cdots\!29}{61\!\cdots\!80}a^{2}+\frac{92\!\cdots\!93}{13\!\cdots\!20}a-\frac{54\!\cdots\!19}{19\!\cdots\!15}$, $\frac{27\!\cdots\!09}{78\!\cdots\!40}a^{11}-\frac{13\!\cdots\!53}{19\!\cdots\!60}a^{10}+\frac{38\!\cdots\!17}{39\!\cdots\!20}a^{9}-\frac{51\!\cdots\!61}{11\!\cdots\!60}a^{8}+\frac{47\!\cdots\!61}{71\!\cdots\!40}a^{7}-\frac{36\!\cdots\!39}{24\!\cdots\!20}a^{6}+\frac{62\!\cdots\!11}{20\!\cdots\!80}a^{5}+\frac{32\!\cdots\!21}{49\!\cdots\!64}a^{4}+\frac{29\!\cdots\!63}{49\!\cdots\!40}a^{3}+\frac{17\!\cdots\!43}{12\!\cdots\!60}a^{2}-\frac{75\!\cdots\!87}{22\!\cdots\!20}a-\frac{21\!\cdots\!59}{19\!\cdots\!15}$
| 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: | \( 37831630.60674724 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 37831630.60674724 \cdot 32}{2\cdot\sqrt{1462520021180890181009408}}\cr\approx \mathstrut & 30.7966404927531 \end{aligned}\] (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 + 34*x^10 - 192*x^9 + 2255*x^8 - 8416*x^7 + 20394*x^6 - 2272*x^5 + 143792*x^4 - 271424*x^3 + 129376*x^2 + 327680*x + 1048576)
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 + 34*x^10 - 192*x^9 + 2255*x^8 - 8416*x^7 + 20394*x^6 - 2272*x^5 + 143792*x^4 - 271424*x^3 + 129376*x^2 + 327680*x + 1048576, 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 + 34*x^10 - 192*x^9 + 2255*x^8 - 8416*x^7 + 20394*x^6 - 2272*x^5 + 143792*x^4 - 271424*x^3 + 129376*x^2 + 327680*x + 1048576);
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 + 34*x^10 - 192*x^9 + 2255*x^8 - 8416*x^7 + 20394*x^6 - 2272*x^5 + 143792*x^4 - 271424*x^3 + 129376*x^2 + 327680*x + 1048576);
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
| \(\Q(\sqrt{-646}) \), 3.1.152.1, 4.0.113509952.4, 6.0.17253512704.1 |
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
| Galois closure: | deg 24 |
| Degree 12 sibling: | 12.2.9621842244611119611904.1 |
| Minimal sibling: | 12.2.9621842244611119611904.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.2.0.1}{2} }^{5}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.2.0.1}{2} }^{5}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | R | R | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.2.0.1}{2} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(19\)
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.152.2t1.b.a | $1$ | $ 2^{3} \cdot 19 $ | \(\Q(\sqrt{-38}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 1.2584.2t1.b.a | $1$ | $ 2^{3} \cdot 17 \cdot 19 $ | \(\Q(\sqrt{-646}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 2.152.3t2.b.a | $2$ | $ 2^{3} \cdot 19 $ | 3.1.152.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.43928.6t3.b.a | $2$ | $ 2^{3} \cdot 17^{2} \cdot 19 $ | 6.2.113509952.3 | $D_{6}$ (as 6T3) | $1$ | $0$ |
| * | 2.43928.4t3.b.a | $2$ | $ 2^{3} \cdot 17^{2} \cdot 19 $ | 4.2.746776.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
| * | 2.43928.12t12.a.b | $2$ | $ 2^{3} \cdot 17^{2} \cdot 19 $ | 12.0.1462520021180890181009408.1 | $D_{12}$ (as 12T12) | $1$ | $0$ |
| * | 2.43928.12t12.a.a | $2$ | $ 2^{3} \cdot 17^{2} \cdot 19 $ | 12.0.1462520021180890181009408.1 | $D_{12}$ (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 *.