Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 9*x^10 + 29*x^9 + 41*x^8 - 82*x^7 - 263*x^6 + 36*x^5 + 705*x^4 + 468*x^3 - 182*x^2 - 169*x - 13)
gp: K = bnfinit(y^12 - 3*y^11 - 9*y^10 + 29*y^9 + 41*y^8 - 82*y^7 - 263*y^6 + 36*y^5 + 705*y^4 + 468*y^3 - 182*y^2 - 169*y - 13, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 - 9*x^10 + 29*x^9 + 41*x^8 - 82*x^7 - 263*x^6 + 36*x^5 + 705*x^4 + 468*x^3 - 182*x^2 - 169*x - 13);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 3*x^11 - 9*x^10 + 29*x^9 + 41*x^8 - 82*x^7 - 263*x^6 + 36*x^5 + 705*x^4 + 468*x^3 - 182*x^2 - 169*x - 13)
\( x^{12} - 3 x^{11} - 9 x^{10} + 29 x^{9} + 41 x^{8} - 82 x^{7} - 263 x^{6} + 36 x^{5} + 705 x^{4} + \cdots - 13 \)
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: | $[8, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(61132828589969773\)
\(\medspace = 7^{8}\cdot 13^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(25.05\) | 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: | $7^{2/3}13^{3/4}\approx 25.052796318948193$ | ||
| Ramified primes: |
\(7\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{702819768927}a^{11}+\frac{32731440254}{702819768927}a^{10}-\frac{3442889489}{234273256309}a^{9}+\frac{20475679100}{234273256309}a^{8}-\frac{77735838932}{702819768927}a^{7}+\frac{300335868640}{702819768927}a^{6}+\frac{311212868260}{702819768927}a^{5}+\frac{27327639675}{234273256309}a^{4}+\frac{69427670132}{702819768927}a^{3}+\frac{269093879063}{702819768927}a^{2}+\frac{88898192902}{234273256309}a+\frac{180759154810}{702819768927}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $9$ | 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: |
$\frac{7465533626}{234273256309}a^{11}-\frac{28261385277}{234273256309}a^{10}-\frac{44922686467}{234273256309}a^{9}+\frac{250155948191}{234273256309}a^{8}+\frac{112995870800}{234273256309}a^{7}-\frac{691568338963}{234273256309}a^{6}-\frac{1443518928668}{234273256309}a^{5}+\frac{1388355057286}{234273256309}a^{4}+\frac{4153714961929}{234273256309}a^{3}+\frac{423933710824}{234273256309}a^{2}-\frac{1575907067709}{234273256309}a-\frac{266980385914}{234273256309}$, $\frac{9134792390}{234273256309}a^{11}-\frac{34756412953}{234273256309}a^{10}-\frac{53759256463}{234273256309}a^{9}+\frac{304922448359}{234273256309}a^{8}+\frac{130533902919}{234273256309}a^{7}-\frac{826941869175}{234273256309}a^{6}-\frac{1771101958537}{234273256309}a^{5}+\frac{1663910746634}{234273256309}a^{4}+\frac{5119366861401}{234273256309}a^{3}+\frac{544301490564}{234273256309}a^{2}-\frac{1946803936103}{234273256309}a-\frac{461497774722}{234273256309}$, $\frac{10576806332}{702819768927}a^{11}-\frac{12743312347}{234273256309}a^{10}-\frac{67077495167}{702819768927}a^{9}+\frac{327192578603}{702819768927}a^{8}+\frac{206546846477}{702819768927}a^{7}-\frac{261170855019}{234273256309}a^{6}-\frac{768456216841}{234273256309}a^{5}+\frac{1084323931037}{702819768927}a^{4}+\frac{6341551614608}{702819768927}a^{3}+\frac{2561815279409}{702819768927}a^{2}-\frac{1141423368436}{702819768927}a-\frac{795657740477}{702819768927}$, $\frac{38842485106}{702819768927}a^{11}-\frac{131640779086}{702819768927}a^{10}-\frac{289344843476}{702819768927}a^{9}+\frac{401485303816}{234273256309}a^{8}+\frac{1068195426317}{702819768927}a^{7}-\frac{3301430374198}{702819768927}a^{6}-\frac{8750132363333}{702819768927}a^{5}+\frac{3973882898912}{702819768927}a^{4}+\frac{23823105787487}{702819768927}a^{3}+\frac{10434173353462}{702819768927}a^{2}-\frac{5325459216209}{702819768927}a-\frac{823894076591}{234273256309}$, $\frac{13216820182}{234273256309}a^{11}-\frac{50720991214}{234273256309}a^{10}-\frac{73668942086}{234273256309}a^{9}+\frac{437083592937}{234273256309}a^{8}+\frac{157283474826}{234273256309}a^{7}-\frac{1151264855003}{234273256309}a^{6}-\frac{2454654067344}{234273256309}a^{5}+\frac{2430148017950}{234273256309}a^{4}+\frac{6800717374100}{234273256309}a^{3}+\frac{501451668900}{234273256309}a^{2}-\frac{2076976936068}{234273256309}a-\frac{450077494517}{234273256309}$, $\frac{20278451660}{234273256309}a^{11}-\frac{200631626678}{702819768927}a^{10}-\frac{488021180596}{702819768927}a^{9}+\frac{1920541785886}{702819768927}a^{8}+\frac{629219023342}{234273256309}a^{7}-\frac{5597073479968}{702819768927}a^{6}-\frac{14074001913325}{702819768927}a^{5}+\frac{6401180883568}{702819768927}a^{4}+\frac{40607568827522}{702819768927}a^{3}+\frac{15080140567430}{702819768927}a^{2}-\frac{14356822799264}{702819768927}a-\frac{4209951974459}{702819768927}$, $\frac{57776022284}{702819768927}a^{11}-\frac{211659987866}{702819768927}a^{10}-\frac{377323317724}{702819768927}a^{9}+\frac{639281087655}{234273256309}a^{8}+\frac{1080424796086}{702819768927}a^{7}-\frac{5379491011424}{702819768927}a^{6}-\frac{11575070306320}{702819768927}a^{5}+\frac{9571800487228}{702819768927}a^{4}+\frac{33792957685510}{702819768927}a^{3}+\frac{4789143199604}{702819768927}a^{2}-\frac{12302153825365}{702819768927}a-\frac{245150853479}{234273256309}$, $\frac{11819794546}{702819768927}a^{11}-\frac{15518072930}{234273256309}a^{10}-\frac{67690564234}{702819768927}a^{9}+\frac{423275265970}{702819768927}a^{8}+\frac{132440765923}{702819768927}a^{7}-\frac{430397483944}{234273256309}a^{6}-\frac{675062711827}{234273256309}a^{5}+\frac{3080741240821}{702819768927}a^{4}+\frac{6119593271179}{702819768927}a^{3}-\frac{1290014146937}{702819768927}a^{2}-\frac{3586297834691}{702819768927}a+\frac{1400356120589}{702819768927}$, $\frac{2409354367}{234273256309}a^{11}-\frac{27669475513}{702819768927}a^{10}-\frac{11321215654}{234273256309}a^{9}+\frac{204241828712}{702819768927}a^{8}+\frac{89246923277}{702819768927}a^{7}-\frac{130607560616}{234273256309}a^{6}-\frac{1520100202166}{702819768927}a^{5}+\frac{270680914448}{234273256309}a^{4}+\frac{2865513651103}{702819768927}a^{3}+\frac{3039954915242}{702819768927}a^{2}-\frac{117482689316}{702819768927}a-\frac{418644674148}{234273256309}$
| 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: | \( 12117.551502516422 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{2}\cdot 12117.551502516422 \cdot 1}{2\cdot\sqrt{61132828589969773}}\cr\approx \mathstrut & 0.247655133802489 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 9*x^10 + 29*x^9 + 41*x^8 - 82*x^7 - 263*x^6 + 36*x^5 + 705*x^4 + 468*x^3 - 182*x^2 - 169*x - 13)
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 - 9*x^10 + 29*x^9 + 41*x^8 - 82*x^7 - 263*x^6 + 36*x^5 + 705*x^4 + 468*x^3 - 182*x^2 - 169*x - 13, 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 - 9*x^10 + 29*x^9 + 41*x^8 - 82*x^7 - 263*x^6 + 36*x^5 + 705*x^4 + 468*x^3 - 182*x^2 - 169*x - 13);
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 - 9*x^10 + 29*x^9 + 41*x^8 - 82*x^7 - 263*x^6 + 36*x^5 + 705*x^4 + 468*x^3 - 182*x^2 - 169*x - 13);
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
$C_4\times A_4$ (as 12T29):
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 $C_4\times A_4$ |
| Character table for $C_4\times A_4$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), \(\Q(\zeta_{7})^+\), 6.6.5274997.1 |
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 16 sibling: | 16.0.134308824412163591281.1 |
| Degree 24 siblings: | deg 24, deg 24 |
| 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 | ${\href{/padicField/2.12.0.1}{12} }$ | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.12.0.1}{12} }$ | R | ${\href{/padicField/11.12.0.1}{12} }$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.12.8.1 | $x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
|
\(13\)
| 13.4.3.1 | $x^{4} + 52$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 13.4.3.1 | $x^{4} + 52$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.91.6t1.j.a | $1$ | $ 7 \cdot 13 $ | 6.6.5274997.1 | $C_6$ (as 6T1) | $0$ | $1$ |
| * | 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.91.6t1.j.b | $1$ | $ 7 \cdot 13 $ | 6.6.5274997.1 | $C_6$ (as 6T1) | $0$ | $1$ |
| * | 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| 1.13.4t1.a.a | $1$ | $ 13 $ | 4.0.2197.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 1.13.4t1.a.b | $1$ | $ 13 $ | 4.0.2197.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 1.91.12t1.a.a | $1$ | $ 7 \cdot 13 $ | 12.0.61132828589969773.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.91.12t1.a.b | $1$ | $ 7 \cdot 13 $ | 12.0.61132828589969773.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.91.12t1.a.c | $1$ | $ 7 \cdot 13 $ | 12.0.61132828589969773.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.91.12t1.a.d | $1$ | $ 7 \cdot 13 $ | 12.0.61132828589969773.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 3.8281.4t4.b.a | $3$ | $ 7^{2} \cdot 13^{2}$ | 4.0.8281.1 | $A_4$ (as 4T4) | $1$ | $-1$ | |
| 3.637.6t6.a.a | $3$ | $ 7^{2} \cdot 13 $ | 6.2.31213.1 | $A_4\times C_2$ (as 6T6) | $1$ | $-1$ | |
| * | 3.107653.12t29.a.a | $3$ | $ 7^{2} \cdot 13^{3}$ | 12.8.61132828589969773.1 | $C_4\times A_4$ (as 12T29) | $0$ | $1$ |
| * | 3.107653.12t29.a.b | $3$ | $ 7^{2} \cdot 13^{3}$ | 12.8.61132828589969773.1 | $C_4\times A_4$ (as 12T29) | $0$ | $1$ |
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 *.