Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 2*x^10 - 13*x^9 + 19*x^8 + 40*x^7 - 108*x^6 - 40*x^5 + 240*x^4 + 64*x^3 - 32*x^2 - 48*x - 16)
gp: K = bnfinit(y^12 - 3*y^11 + 2*y^10 - 13*y^9 + 19*y^8 + 40*y^7 - 108*y^6 - 40*y^5 + 240*y^4 + 64*y^3 - 32*y^2 - 48*y - 16, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 + 2*x^10 - 13*x^9 + 19*x^8 + 40*x^7 - 108*x^6 - 40*x^5 + 240*x^4 + 64*x^3 - 32*x^2 - 48*x - 16);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 3*x^11 + 2*x^10 - 13*x^9 + 19*x^8 + 40*x^7 - 108*x^6 - 40*x^5 + 240*x^4 + 64*x^3 - 32*x^2 - 48*x - 16)
\( x^{12} - 3 x^{11} + 2 x^{10} - 13 x^{9} + 19 x^{8} + 40 x^{7} - 108 x^{6} - 40 x^{5} + 240 x^{4} + \cdots - 16 \)
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: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(4492085992834072576\)
\(\medspace = 2^{17}\cdot 17^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(35.84\) | 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: | $2^{8/3}17^{11/12}\approx 85.24289022322928$ | ||
| Ramified primes: |
\(2\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{34}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{24}a^{9}-\frac{1}{24}a^{8}+\frac{1}{6}a^{7}-\frac{1}{24}a^{6}-\frac{11}{24}a^{5}+\frac{5}{12}a^{4}+\frac{1}{6}a^{3}+\frac{1}{6}a^{2}-\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{48}a^{10}-\frac{1}{48}a^{9}-\frac{1}{24}a^{8}-\frac{7}{48}a^{7}+\frac{1}{48}a^{6}-\frac{1}{6}a^{5}+\frac{5}{24}a^{4}+\frac{1}{12}a^{3}+\frac{5}{12}a^{2}-\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{20193072}a^{11}-\frac{65583}{6731024}a^{10}+\frac{50185}{3365512}a^{9}+\frac{26095}{6731024}a^{8}-\frac{1120145}{6731024}a^{7}+\frac{94940}{1262067}a^{6}-\frac{1685729}{10096536}a^{5}-\frac{336625}{5048268}a^{4}-\frac{797261}{1682756}a^{3}+\frac{490162}{1262067}a^{2}-\frac{277811}{2524134}a+\frac{100894}{420689}$
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
$C_{2}$, which has order $2$
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: | $7$ | 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: |
$\frac{246007}{10096536}a^{11}-\frac{117638}{1262067}a^{10}+\frac{272507}{2524134}a^{9}-\frac{428656}{1262067}a^{8}+\frac{3341461}{5048268}a^{7}+\frac{1228749}{1682756}a^{6}-\frac{38725357}{10096536}a^{5}+\frac{4730165}{2524134}a^{4}+\frac{5204265}{841378}a^{3}-\frac{1587195}{420689}a^{2}-\frac{1259053}{2524134}a-\frac{545141}{1262067}$, $\frac{87461}{1262067}a^{11}-\frac{1267799}{5048268}a^{10}+\frac{1421929}{5048268}a^{9}-\frac{2603003}{2524134}a^{8}+\frac{9684877}{5048268}a^{7}+\frac{2966253}{1682756}a^{6}-\frac{22439521}{2524134}a^{5}+\frac{5829143}{2524134}a^{4}+\frac{6981429}{420689}a^{3}-\frac{4811317}{841378}a^{2}-\frac{2511896}{1262067}a-\frac{3110462}{1262067}$, $\frac{987013}{10096536}a^{11}-\frac{2936959}{10096536}a^{10}+\frac{545645}{3365512}a^{9}-\frac{1972651}{1682756}a^{8}+\frac{5828857}{3365512}a^{7}+\frac{43521071}{10096536}a^{6}-\frac{37497237}{3365512}a^{5}-\frac{8427309}{1682756}a^{4}+\frac{34190795}{1262067}a^{3}+\frac{16085113}{2524134}a^{2}-\frac{24984227}{2524134}a-\frac{4131724}{1262067}$, $\frac{564903}{6731024}a^{11}-\frac{3454799}{20193072}a^{10}-\frac{136079}{2524134}a^{9}-\frac{20064803}{20193072}a^{8}+\frac{11044079}{20193072}a^{7}+\frac{46770419}{10096536}a^{6}-\frac{54046585}{10096536}a^{5}-\frac{54057221}{5048268}a^{4}+\frac{78449017}{5048268}a^{3}+\frac{28998224}{1262067}a^{2}+\frac{23802667}{2524134}a+\frac{1132686}{420689}$, $\frac{459889}{10096536}a^{11}-\frac{271601}{1262067}a^{10}+\frac{3606239}{10096536}a^{9}-\frac{8568035}{10096536}a^{8}+\frac{9902443}{5048268}a^{7}-\frac{18835}{3365512}a^{6}-\frac{9363017}{1262067}a^{5}+\frac{39227803}{5048268}a^{4}+\frac{3963506}{420689}a^{3}-\frac{7424600}{420689}a^{2}+\frac{5653900}{1262067}a+\frac{5512645}{1262067}$, $\frac{73751}{841378}a^{11}-\frac{256111}{1262067}a^{10}+\frac{86045}{1682756}a^{9}-\frac{1981047}{1682756}a^{8}+\frac{861995}{841378}a^{7}+\frac{6262881}{1682756}a^{6}-\frac{29179211}{5048268}a^{5}-\frac{11117666}{1262067}a^{4}+\frac{38067641}{2524134}a^{3}+\frac{14310253}{841378}a^{2}+\frac{11506964}{1262067}a+\frac{2755978}{1262067}$, $\frac{3594565}{20193072}a^{11}-\frac{10998175}{20193072}a^{10}+\frac{1650265}{5048268}a^{9}-\frac{43785487}{20193072}a^{8}+\frac{68269303}{20193072}a^{7}+\frac{78115721}{10096536}a^{6}-\frac{207278755}{10096536}a^{5}-\frac{39789593}{5048268}a^{4}+\frac{249592151}{5048268}a^{3}+\frac{13899692}{1262067}a^{2}-\frac{14311707}{841378}a-\frac{2580278}{420689}$
| 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: | \( 187300.206044 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 187300.206044 \cdot 2}{2\cdot\sqrt{4492085992834072576}}\cr\approx \mathstrut & 2.20370613388 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 2*x^10 - 13*x^9 + 19*x^8 + 40*x^7 - 108*x^6 - 40*x^5 + 240*x^4 + 64*x^3 - 32*x^2 - 48*x - 16)
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 - 3*x^11 + 2*x^10 - 13*x^9 + 19*x^8 + 40*x^7 - 108*x^6 - 40*x^5 + 240*x^4 + 64*x^3 - 32*x^2 - 48*x - 16, 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 - 3*x^11 + 2*x^10 - 13*x^9 + 19*x^8 + 40*x^7 - 108*x^6 - 40*x^5 + 240*x^4 + 64*x^3 - 32*x^2 - 48*x - 16);
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 - 3*x^11 + 2*x^10 - 13*x^9 + 19*x^8 + 40*x^7 - 108*x^6 - 40*x^5 + 240*x^4 + 64*x^3 - 32*x^2 - 48*x - 16);
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
$S_4^2:D_4$ (as 12T260):
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 4608 |
| The 65 conjugacy class representatives for $S_4^2:D_4$ |
| Character table for $S_4^2:D_4$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 6.4.45435424.1 |
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 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.2.2246042996417036288.3 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{8}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.6.11.7 | $x^{6} + 4 x^{2} + 4 x + 2$ | $6$ | $1$ | $11$ | $S_4\times C_2$ | $[8/3, 8/3, 3]_{3}^{2}$ | |
|
\(17\)
| 17.12.11.2 | $x^{12} + 34$ | $12$ | $1$ | $11$ | $S_3 \times C_4$ | $[\ ]_{12}^{2}$ |