Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 42*x^10 + 43*x^9 + 811*x^8 - 349*x^7 - 8317*x^6 + 2578*x^5 + 56193*x^4 - 7783*x^3 - 209160*x^2 + 78428*x + 578536)
gp: K = bnfinit(y^12 - 2*y^11 - 42*y^10 + 43*y^9 + 811*y^8 - 349*y^7 - 8317*y^6 + 2578*y^5 + 56193*y^4 - 7783*y^3 - 209160*y^2 + 78428*y + 578536, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 - 42*x^10 + 43*x^9 + 811*x^8 - 349*x^7 - 8317*x^6 + 2578*x^5 + 56193*x^4 - 7783*x^3 - 209160*x^2 + 78428*x + 578536);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 - 42*x^10 + 43*x^9 + 811*x^8 - 349*x^7 - 8317*x^6 + 2578*x^5 + 56193*x^4 - 7783*x^3 - 209160*x^2 + 78428*x + 578536)
\( x^{12} - 2 x^{11} - 42 x^{10} + 43 x^{9} + 811 x^{8} - 349 x^{7} - 8317 x^{6} + 2578 x^{5} + \cdots + 578536 \)
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: |
\(125049841902709607569\)
\(\medspace = 7^{8}\cdot 167^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(47.29\) | 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: | $7^{2/3}167^{1/2}\approx 47.28865141512203$ | ||
| Ramified primes: |
\(7\), \(167\)
| 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: | \(\Q(\sqrt{-167}) \), 6.0.11182568663.3$^{3}$, 8.0.1867488966721.2$^{4}$, 12.0.125049841902709607569.1$^{12}$, deg 24$^{12}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{42}a^{9}+\frac{3}{14}a^{8}-\frac{3}{14}a^{7}-\frac{1}{6}a^{6}+\frac{17}{42}a^{5}+\frac{5}{14}a^{4}+\frac{1}{42}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{42}a^{10}-\frac{1}{7}a^{8}-\frac{5}{21}a^{7}-\frac{2}{21}a^{6}-\frac{2}{7}a^{5}-\frac{4}{21}a^{4}+\frac{5}{42}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{63\!\cdots\!92}a^{11}+\frac{39\!\cdots\!65}{15\!\cdots\!23}a^{10}+\frac{57\!\cdots\!45}{31\!\cdots\!46}a^{9}-\frac{11\!\cdots\!71}{63\!\cdots\!92}a^{8}+\frac{71\!\cdots\!85}{63\!\cdots\!92}a^{7}-\frac{25\!\cdots\!67}{63\!\cdots\!92}a^{6}-\frac{27\!\cdots\!53}{21\!\cdots\!64}a^{5}+\frac{19\!\cdots\!43}{76\!\cdots\!63}a^{4}+\frac{31\!\cdots\!93}{63\!\cdots\!92}a^{3}+\frac{44\!\cdots\!57}{91\!\cdots\!56}a^{2}-\frac{21\!\cdots\!91}{76\!\cdots\!63}a-\frac{84\!\cdots\!73}{22\!\cdots\!89}$
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
$C_{66}$, which has order $66$ (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: |
$\frac{1300863933856}{70\!\cdots\!73}a^{11}-\frac{15444921535895}{14\!\cdots\!46}a^{10}-\frac{196839193387307}{42\!\cdots\!38}a^{9}+\frac{424933635359035}{14\!\cdots\!46}a^{8}+\frac{476588192492201}{70\!\cdots\!73}a^{7}-\frac{83\!\cdots\!78}{21\!\cdots\!19}a^{6}-\frac{11\!\cdots\!01}{21\!\cdots\!19}a^{5}+\frac{21\!\cdots\!49}{70\!\cdots\!73}a^{4}+\frac{12\!\cdots\!25}{42\!\cdots\!38}a^{3}-\frac{76\!\cdots\!21}{60\!\cdots\!34}a^{2}-\frac{47\!\cdots\!45}{60\!\cdots\!34}a+\frac{10\!\cdots\!61}{30\!\cdots\!17}$, $\frac{1300863933856}{70\!\cdots\!73}a^{11}-\frac{15444921535895}{14\!\cdots\!46}a^{10}-\frac{196839193387307}{42\!\cdots\!38}a^{9}+\frac{424933635359035}{14\!\cdots\!46}a^{8}+\frac{476588192492201}{70\!\cdots\!73}a^{7}-\frac{83\!\cdots\!78}{21\!\cdots\!19}a^{6}-\frac{11\!\cdots\!01}{21\!\cdots\!19}a^{5}+\frac{21\!\cdots\!49}{70\!\cdots\!73}a^{4}+\frac{12\!\cdots\!25}{42\!\cdots\!38}a^{3}-\frac{76\!\cdots\!21}{60\!\cdots\!34}a^{2}-\frac{47\!\cdots\!45}{60\!\cdots\!34}a+\frac{72\!\cdots\!44}{30\!\cdots\!17}$, $\frac{25\!\cdots\!83}{63\!\cdots\!92}a^{11}-\frac{30\!\cdots\!00}{15\!\cdots\!23}a^{10}-\frac{29\!\cdots\!05}{31\!\cdots\!46}a^{9}+\frac{25\!\cdots\!71}{63\!\cdots\!92}a^{8}+\frac{23\!\cdots\!35}{21\!\cdots\!64}a^{7}-\frac{77\!\cdots\!85}{21\!\cdots\!64}a^{6}-\frac{11\!\cdots\!61}{21\!\cdots\!64}a^{5}+\frac{71\!\cdots\!13}{31\!\cdots\!46}a^{4}+\frac{39\!\cdots\!71}{21\!\cdots\!64}a^{3}-\frac{23\!\cdots\!81}{30\!\cdots\!52}a^{2}+\frac{72\!\cdots\!17}{45\!\cdots\!78}a+\frac{22\!\cdots\!21}{22\!\cdots\!89}$, $\frac{16\!\cdots\!87}{31\!\cdots\!46}a^{11}-\frac{11\!\cdots\!09}{31\!\cdots\!46}a^{10}-\frac{31\!\cdots\!62}{53\!\cdots\!41}a^{9}+\frac{13\!\cdots\!29}{15\!\cdots\!23}a^{8}+\frac{12\!\cdots\!43}{15\!\cdots\!26}a^{7}-\frac{26\!\cdots\!33}{31\!\cdots\!46}a^{6}+\frac{15\!\cdots\!81}{31\!\cdots\!46}a^{5}+\frac{71\!\cdots\!39}{15\!\cdots\!23}a^{4}-\frac{50\!\cdots\!91}{15\!\cdots\!23}a^{3}-\frac{62\!\cdots\!37}{45\!\cdots\!78}a^{2}+\frac{63\!\cdots\!55}{45\!\cdots\!78}a+\frac{17\!\cdots\!72}{76\!\cdots\!63}$, $\frac{43\!\cdots\!92}{15\!\cdots\!23}a^{11}+\frac{18\!\cdots\!67}{15\!\cdots\!23}a^{10}-\frac{35\!\cdots\!53}{31\!\cdots\!46}a^{9}-\frac{15\!\cdots\!85}{31\!\cdots\!46}a^{8}-\frac{50\!\cdots\!05}{31\!\cdots\!46}a^{7}+\frac{86\!\cdots\!01}{10\!\cdots\!82}a^{6}+\frac{22\!\cdots\!69}{31\!\cdots\!46}a^{5}-\frac{69\!\cdots\!99}{10\!\cdots\!82}a^{4}-\frac{21\!\cdots\!85}{31\!\cdots\!46}a^{3}+\frac{63\!\cdots\!91}{22\!\cdots\!89}a^{2}+\frac{85\!\cdots\!02}{22\!\cdots\!89}a-\frac{27\!\cdots\!89}{22\!\cdots\!89}$
| 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: | \( 2950.17006547 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 2950.17006547 \cdot 66}{2\cdot\sqrt{125049841902709607569}}\cr\approx \mathstrut & 0.535671608401 \end{aligned}\] (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 - 42*x^10 + 43*x^9 + 811*x^8 - 349*x^7 - 8317*x^6 + 2578*x^5 + 56193*x^4 - 7783*x^3 - 209160*x^2 + 78428*x + 578536)
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 - 42*x^10 + 43*x^9 + 811*x^8 - 349*x^7 - 8317*x^6 + 2578*x^5 + 56193*x^4 - 7783*x^3 - 209160*x^2 + 78428*x + 578536, 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 - 42*x^10 + 43*x^9 + 811*x^8 - 349*x^7 - 8317*x^6 + 2578*x^5 + 56193*x^4 - 7783*x^3 - 209160*x^2 + 78428*x + 578536);
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 - 42*x^10 + 43*x^9 + 811*x^8 - 349*x^7 - 8317*x^6 + 2578*x^5 + 56193*x^4 - 7783*x^3 - 209160*x^2 + 78428*x + 578536);
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
$C_2\times A_4$ (as 12T7):
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 8 conjugacy class representatives for $A_4 \times C_2$ |
| Character table for $A_4 \times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-167}) \), \(\Q(\zeta_{7})^+\), 6.0.400967.1, 6.0.11182568663.3, 6.6.66961489.1 |
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 6 sibling: | 6.0.400967.1 |
| Degree 8 sibling: | 8.0.1867488966721.2 |
| Degree 12 sibling: | 12.0.4483841009097121.1 |
| Minimal sibling: | 6.0.400967.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
|
\(167\)
| 167.2.1.2 | $x^{2} + 167$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 167.2.1.2 | $x^{2} + 167$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |