Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 4*x^10 + 14*x^9 - 15*x^8 + 90*x^7 + 50*x^6 - 644*x^5 + 1098*x^4 - 1152*x^3 + 822*x^2 - 230*x + 25)
gp: K = bnfinit(y^12 - 2*y^11 - 4*y^10 + 14*y^9 - 15*y^8 + 90*y^7 + 50*y^6 - 644*y^5 + 1098*y^4 - 1152*y^3 + 822*y^2 - 230*y + 25, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 - 4*x^10 + 14*x^9 - 15*x^8 + 90*x^7 + 50*x^6 - 644*x^5 + 1098*x^4 - 1152*x^3 + 822*x^2 - 230*x + 25);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 - 4*x^10 + 14*x^9 - 15*x^8 + 90*x^7 + 50*x^6 - 644*x^5 + 1098*x^4 - 1152*x^3 + 822*x^2 - 230*x + 25)
\( x^{12} - 2 x^{11} - 4 x^{10} + 14 x^{9} - 15 x^{8} + 90 x^{7} + 50 x^{6} - 644 x^{5} + 1098 x^{4} + \cdots + 25 \)
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: |
\(36138776487264256\)
\(\medspace = 2^{18}\cdot 13^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.98\) | 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: | $2^{7/4}13^{5/6}\approx 28.516000872850142$ | ||
| Ramified primes: |
\(2\), \(13\)
| 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 not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}+\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a$, $\frac{1}{5}a^{8}-\frac{2}{5}a^{6}-\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{5}a^{9}-\frac{1}{5}a^{5}+\frac{1}{5}a$, $\frac{1}{25}a^{10}+\frac{2}{25}a^{9}+\frac{2}{25}a^{8}-\frac{2}{25}a^{7}+\frac{8}{25}a^{6}+\frac{11}{25}a^{5}-\frac{7}{25}a^{4}-\frac{4}{25}a^{3}+\frac{1}{25}a^{2}$, $\frac{1}{809295713075}a^{11}+\frac{5044346294}{809295713075}a^{10}+\frac{43775923621}{809295713075}a^{9}-\frac{53079321803}{809295713075}a^{8}+\frac{4055747824}{809295713075}a^{7}-\frac{97873165508}{809295713075}a^{6}-\frac{75441683782}{161859142615}a^{5}+\frac{154480756492}{809295713075}a^{4}+\frac{300903932873}{809295713075}a^{3}-\frac{149647336298}{809295713075}a^{2}+\frac{56143815648}{161859142615}a+\frac{14787162529}{32371828523}$
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
$C_{2}$, 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: | $5$ | 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: |
$\frac{8546279862}{809295713075}a^{11}-\frac{9038068074}{809295713075}a^{10}-\frac{44001671557}{809295713075}a^{9}+\frac{16543442541}{161859142615}a^{8}-\frac{46413825058}{809295713075}a^{7}+\frac{698216817223}{809295713075}a^{6}+\frac{1119245215373}{809295713075}a^{5}-\frac{4561221533187}{809295713075}a^{4}+\frac{5144459448579}{809295713075}a^{3}-\frac{3883916277748}{809295713075}a^{2}+\frac{222456736243}{161859142615}a+\frac{47779296045}{32371828523}$, $\frac{3801026094}{809295713075}a^{11}+\frac{1192340129}{809295713075}a^{10}-\frac{3786951118}{161859142615}a^{9}+\frac{8361324594}{809295713075}a^{8}-\frac{611203539}{161859142615}a^{7}+\frac{318331764022}{809295713075}a^{6}+\frac{902956489893}{809295713075}a^{5}-\frac{857262684393}{809295713075}a^{4}+\frac{144466762422}{161859142615}a^{3}-\frac{838643973494}{809295713075}a^{2}+\frac{55696890579}{161859142615}a-\frac{15779173565}{32371828523}$, $\frac{339850680}{32371828523}a^{11}-\frac{30649460953}{809295713075}a^{10}-\frac{33236759076}{809295713075}a^{9}+\frac{188771166099}{809295713075}a^{8}-\frac{185376187094}{809295713075}a^{7}+\frac{787272881916}{809295713075}a^{6}-\frac{688540643848}{809295713075}a^{5}-\frac{8351973850349}{809295713075}a^{4}+\frac{13579477043942}{809295713075}a^{3}-\frac{13733144453458}{809295713075}a^{2}+\frac{2146482003678}{161859142615}a-\frac{58148266155}{32371828523}$, $\frac{32132926338}{809295713075}a^{11}-\frac{68031415942}{809295713075}a^{10}-\frac{26777848219}{161859142615}a^{9}+\frac{483782931328}{809295713075}a^{8}-\frac{95217795229}{161859142615}a^{7}+\frac{2792295471634}{809295713075}a^{6}+\frac{1403787691076}{809295713075}a^{5}-\frac{21936354496081}{809295713075}a^{4}+\frac{7281438484739}{161859142615}a^{3}-\frac{33870261237273}{809295713075}a^{2}+\frac{3827741917938}{161859142615}a-\frac{71916300510}{32371828523}$, $\frac{42670983069}{809295713075}a^{11}-\frac{50540368668}{809295713075}a^{10}-\frac{232126441119}{809295713075}a^{9}+\frac{88132121337}{161859142615}a^{8}-\frac{184873306761}{809295713075}a^{7}+\frac{3439089085666}{809295713075}a^{6}+\frac{5139558302776}{809295713075}a^{5}-\frac{25007931870499}{809295713075}a^{4}+\frac{24729210939128}{809295713075}a^{3}-\frac{16351506102341}{809295713075}a^{2}+\frac{178938781677}{32371828523}a+\frac{368171905628}{32371828523}$
| 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: | \( 981.852738997 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 981.852738997 \cdot 2}{2\cdot\sqrt{36138776487264256}}\cr\approx \mathstrut & 0.317788987725 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 4*x^10 + 14*x^9 - 15*x^8 + 90*x^7 + 50*x^6 - 644*x^5 + 1098*x^4 - 1152*x^3 + 822*x^2 - 230*x + 25)
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 - 4*x^10 + 14*x^9 - 15*x^8 + 90*x^7 + 50*x^6 - 644*x^5 + 1098*x^4 - 1152*x^3 + 822*x^2 - 230*x + 25, 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 - 4*x^10 + 14*x^9 - 15*x^8 + 90*x^7 + 50*x^6 - 644*x^5 + 1098*x^4 - 1152*x^3 + 822*x^2 - 230*x + 25);
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 - 4*x^10 + 14*x^9 - 15*x^8 + 90*x^7 + 50*x^6 - 644*x^5 + 1098*x^4 - 1152*x^3 + 822*x^2 - 230*x + 25);
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
| \(\Q(\sqrt{-13}) \), 3.3.169.1, 6.0.23762752.1, 6.4.23762752.1, 6.2.1827904.1 |
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: | 6.4.23762752.1 |
| Degree 8 sibling: | 8.0.79082438656.1 |
| Degree 12 sibling: | 12.4.36138776487264256.2 |
| Minimal sibling: | 6.4.23762752.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{6}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.1.0.1}{1} }^{12}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.18.49 | $x^{12} + 2 x^{11} - 4 x^{10} + 12 x^{9} + 78 x^{8} + 96 x^{7} + 16 x^{6} + 72 x^{5} + 180 x^{4} + 248 x^{3} + 248$ | $4$ | $3$ | $18$ | $A_4 \times C_2$ | $[2, 2, 2]^{3}$ |
|
\(13\)
| 13.6.5.2 | $x^{6} + 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 13.6.5.2 | $x^{6} + 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |