Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 15*x^10 + 64*x^9 + 177*x^8 - 584*x^7 - 677*x^6 + 1500*x^5 + 2178*x^4 - 1040*x^3 + 18544*x^2 - 5264*x + 33856)
gp: K = bnfinit(y^12 - 4*y^11 - 15*y^10 + 64*y^9 + 177*y^8 - 584*y^7 - 677*y^6 + 1500*y^5 + 2178*y^4 - 1040*y^3 + 18544*y^2 - 5264*y + 33856, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 15*x^10 + 64*x^9 + 177*x^8 - 584*x^7 - 677*x^6 + 1500*x^5 + 2178*x^4 - 1040*x^3 + 18544*x^2 - 5264*x + 33856);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 - 15*x^10 + 64*x^9 + 177*x^8 - 584*x^7 - 677*x^6 + 1500*x^5 + 2178*x^4 - 1040*x^3 + 18544*x^2 - 5264*x + 33856)
\( x^{12} - 4 x^{11} - 15 x^{10} + 64 x^{9} + 177 x^{8} - 584 x^{7} - 677 x^{6} + 1500 x^{5} + 2178 x^{4} + \cdots + 33856 \)
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: |
\(4681545978686741527104\)
\(\medspace = 2^{6}\cdot 3^{6}\cdot 7^{6}\cdot 31^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(63.95\) | 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: | $2^{3/2}3^{1/2}7^{1/2}31^{2/3}\approx 127.9074291730327$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{32}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{8}a^{7}-\frac{1}{16}a^{6}+\frac{1}{16}a^{5}-\frac{1}{4}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{15\!\cdots\!68}a^{11}-\frac{674447246283747}{15\!\cdots\!68}a^{10}+\frac{10\!\cdots\!75}{38\!\cdots\!92}a^{9}+\frac{18\!\cdots\!01}{77\!\cdots\!84}a^{8}-\frac{882341773288893}{15\!\cdots\!68}a^{7}+\frac{38\!\cdots\!43}{15\!\cdots\!68}a^{6}+\frac{25\!\cdots\!89}{38\!\cdots\!92}a^{5}+\frac{14\!\cdots\!61}{77\!\cdots\!84}a^{4}-\frac{16\!\cdots\!63}{38\!\cdots\!92}a^{3}+\frac{54\!\cdots\!39}{19\!\cdots\!96}a^{2}-\frac{43\!\cdots\!53}{96\!\cdots\!98}a-\frac{133483396720884}{48\!\cdots\!49}$
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
$C_{1554}$, which has order $1554$ (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: |
$\frac{4242633945061}{77\!\cdots\!84}a^{11}-\frac{6969177912133}{38\!\cdots\!92}a^{10}-\frac{70927512294851}{77\!\cdots\!84}a^{9}+\frac{76796153832731}{38\!\cdots\!92}a^{8}+\frac{10\!\cdots\!93}{77\!\cdots\!84}a^{7}-\frac{823573627469757}{38\!\cdots\!92}a^{6}-\frac{39\!\cdots\!93}{77\!\cdots\!84}a^{5}+\frac{894863143807477}{38\!\cdots\!92}a^{4}+\frac{63\!\cdots\!89}{38\!\cdots\!92}a^{3}-\frac{49\!\cdots\!33}{19\!\cdots\!96}a^{2}+\frac{722541436425786}{48\!\cdots\!49}a-\frac{46\!\cdots\!45}{48\!\cdots\!49}$, $\frac{163914309540}{48\!\cdots\!49}a^{11}+\frac{3444487414419}{77\!\cdots\!84}a^{10}+\frac{54407319459527}{77\!\cdots\!84}a^{9}+\frac{26257659323973}{38\!\cdots\!92}a^{8}-\frac{316248583783401}{38\!\cdots\!92}a^{7}-\frac{841520568582139}{77\!\cdots\!84}a^{6}+\frac{971850994696437}{77\!\cdots\!84}a^{5}+\frac{40\!\cdots\!91}{38\!\cdots\!92}a^{4}+\frac{66\!\cdots\!59}{38\!\cdots\!92}a^{3}+\frac{31\!\cdots\!67}{96\!\cdots\!98}a^{2}+\frac{16\!\cdots\!63}{96\!\cdots\!98}a+\frac{66\!\cdots\!95}{48\!\cdots\!49}$, $\frac{3029345863431}{77\!\cdots\!84}a^{11}+\frac{25752921411015}{77\!\cdots\!84}a^{10}-\frac{14718337680905}{38\!\cdots\!92}a^{9}-\frac{163636832527285}{38\!\cdots\!92}a^{8}+\frac{10\!\cdots\!83}{77\!\cdots\!84}a^{7}+\frac{17\!\cdots\!13}{77\!\cdots\!84}a^{6}-\frac{62\!\cdots\!23}{38\!\cdots\!92}a^{5}+\frac{41\!\cdots\!75}{38\!\cdots\!92}a^{4}+\frac{28\!\cdots\!43}{48\!\cdots\!49}a^{3}-\frac{78\!\cdots\!75}{96\!\cdots\!98}a^{2}+\frac{43\!\cdots\!69}{48\!\cdots\!49}a+\frac{41\!\cdots\!57}{48\!\cdots\!49}$, $\frac{12842698629637}{77\!\cdots\!84}a^{11}+\frac{29015751867341}{77\!\cdots\!84}a^{10}+\frac{159363263228857}{38\!\cdots\!92}a^{9}-\frac{313766795391367}{38\!\cdots\!92}a^{8}-\frac{43\!\cdots\!95}{77\!\cdots\!84}a^{7}+\frac{62\!\cdots\!39}{77\!\cdots\!84}a^{6}+\frac{14\!\cdots\!95}{38\!\cdots\!92}a^{5}-\frac{16\!\cdots\!27}{38\!\cdots\!92}a^{4}-\frac{63\!\cdots\!45}{48\!\cdots\!49}a^{3}+\frac{11\!\cdots\!89}{96\!\cdots\!98}a^{2}-\frac{71\!\cdots\!91}{48\!\cdots\!49}a-\frac{35\!\cdots\!01}{48\!\cdots\!49}$, $\frac{97584860821}{38\!\cdots\!92}a^{11}-\frac{283652498611}{48\!\cdots\!49}a^{10}-\frac{2535937923353}{38\!\cdots\!92}a^{9}+\frac{18343575066774}{48\!\cdots\!49}a^{8}-\frac{560376006394363}{38\!\cdots\!92}a^{7}-\frac{4545094894552}{48\!\cdots\!49}a^{6}+\frac{27\!\cdots\!81}{38\!\cdots\!92}a^{5}+\frac{132016905405990}{48\!\cdots\!49}a^{4}-\frac{90\!\cdots\!49}{19\!\cdots\!96}a^{3}+\frac{35\!\cdots\!54}{48\!\cdots\!49}a^{2}-\frac{32\!\cdots\!19}{48\!\cdots\!49}a-\frac{14\!\cdots\!15}{48\!\cdots\!49}$
| 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: | \( 25646.03638896619 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 25646.03638896619 \cdot 1554}{2\cdot\sqrt{4681545978686741527104}}\cr\approx \mathstrut & 17.9194978957794 \end{aligned}\] (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 - 15*x^10 + 64*x^9 + 177*x^8 - 584*x^7 - 677*x^6 + 1500*x^5 + 2178*x^4 - 1040*x^3 + 18544*x^2 - 5264*x + 33856)
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 - 15*x^10 + 64*x^9 + 177*x^8 - 584*x^7 - 677*x^6 + 1500*x^5 + 2178*x^4 - 1040*x^3 + 18544*x^2 - 5264*x + 33856, 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 - 15*x^10 + 64*x^9 + 177*x^8 - 584*x^7 - 677*x^6 + 1500*x^5 + 2178*x^4 - 1040*x^3 + 18544*x^2 - 5264*x + 33856);
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 - 15*x^10 + 64*x^9 + 177*x^8 - 584*x^7 - 677*x^6 + 1500*x^5 + 2178*x^4 - 1040*x^3 + 18544*x^2 - 5264*x + 33856);
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
$C_2^2\times A_4$ (as 12T25):
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 48 |
| The 16 conjugacy class representatives for $C_2^2 \times A_4$ |
| Character table for $C_2^2 \times A_4$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), 3.3.961.1, 6.6.199480536.1, 6.0.68421823848.1, 6.0.316767703.1 |
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 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| 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 | R | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.2.3.3 | $x^{2} + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.2.3.3 | $x^{2} + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
|
\(3\)
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(7\)
| 7.12.6.1 | $x^{12} + 44 x^{10} + 10 x^{9} + 786 x^{8} + 22 x^{7} + 6899 x^{6} - 3434 x^{5} + 31050 x^{4} - 28440 x^{3} + 84557 x^{2} - 48082 x + 107648$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(31\)
| 31.6.4.1 | $x^{6} + 87 x^{5} + 2532 x^{4} + 24973 x^{3} + 10293 x^{2} + 78438 x + 748956$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 31.6.4.1 | $x^{6} + 87 x^{5} + 2532 x^{4} + 24973 x^{3} + 10293 x^{2} + 78438 x + 748956$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |