Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 + 6*x^8 - 22*x^7 + 48*x^6 - 48*x^5 + 104*x^4 - 132*x^3 - 144*x^2 + 112*x + 144)
gp: K = bnfinit(y^10 - 3*y^9 + 6*y^8 - 22*y^7 + 48*y^6 - 48*y^5 + 104*y^4 - 132*y^3 - 144*y^2 + 112*y + 144, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 3*x^9 + 6*x^8 - 22*x^7 + 48*x^6 - 48*x^5 + 104*x^4 - 132*x^3 - 144*x^2 + 112*x + 144);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 3*x^9 + 6*x^8 - 22*x^7 + 48*x^6 - 48*x^5 + 104*x^4 - 132*x^3 - 144*x^2 + 112*x + 144)
\( x^{10} - 3x^{9} + 6x^{8} - 22x^{7} + 48x^{6} - 48x^{5} + 104x^{4} - 132x^{3} - 144x^{2} + 112x + 144 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(810091364106816\)
\(\medspace = 2^{6}\cdot 3^{10}\cdot 11^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.96\) | 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: | $2^{2/3}3^{7/6}11^{4/5}\approx 38.94415416349148$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{44}a^{8}+\frac{5}{44}a^{7}-\frac{3}{44}a^{6}-\frac{5}{44}a^{5}-\frac{5}{22}a^{4}+\frac{9}{22}a^{3}-\frac{5}{22}a^{2}-\frac{4}{11}a+\frac{5}{11}$, $\frac{1}{129624}a^{9}-\frac{865}{129624}a^{8}-\frac{331}{16203}a^{7}-\frac{149}{32406}a^{6}+\frac{2071}{64812}a^{5}-\frac{5867}{32406}a^{4}+\frac{9409}{32406}a^{3}+\frac{5029}{16203}a^{2}+\frac{2969}{16203}a+\frac{2234}{5401}$
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
$C_{10}$, which has order $10$
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: |
$\frac{133}{21604}a^{9}-\frac{151}{21604}a^{8}+\frac{845}{21604}a^{7}-\frac{668}{5401}a^{6}+\frac{311}{1964}a^{5}-\frac{1697}{10802}a^{4}+\frac{9987}{10802}a^{3}-\frac{31}{10802}a^{2}-\frac{4684}{5401}a-\frac{3035}{5401}$, $\frac{4183}{129624}a^{9}-\frac{6499}{129624}a^{8}+\frac{7553}{64812}a^{7}-\frac{37201}{64812}a^{6}+\frac{54781}{64812}a^{5}-\frac{7369}{16203}a^{4}+\frac{9041}{2946}a^{3}-\frac{21701}{16203}a^{2}-\frac{86443}{16203}a-\frac{19039}{5401}$, $\frac{205}{64812}a^{9}-\frac{565}{64812}a^{8}+\frac{697}{64812}a^{7}-\frac{1085}{16203}a^{6}+\frac{13919}{64812}a^{5}-\frac{61}{32406}a^{4}+\frac{20525}{32406}a^{3}-\frac{16823}{32406}a^{2}-\frac{24449}{16203}a-\frac{4685}{5401}$, $\frac{1637}{21604}a^{9}-\frac{604}{5401}a^{8}+\frac{1416}{5401}a^{7}-\frac{13029}{10802}a^{6}+\frac{36569}{21604}a^{5}-\frac{549}{982}a^{4}+\frac{32760}{5401}a^{3}+\frac{1959}{10802}a^{2}-\frac{6456}{491}a-\frac{47087}{5401}$, $\frac{413}{129624}a^{9}-\frac{779}{129624}a^{8}+\frac{2045}{32406}a^{7}-\frac{9653}{64812}a^{6}+\frac{12767}{64812}a^{5}-\frac{4412}{16203}a^{4}+\frac{6700}{16203}a^{3}+\frac{19196}{16203}a^{2}-\frac{5231}{16203}a-\frac{929}{5401}$
| 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: | \( 3791.68777699 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 3791.68777699 \cdot 10}{2\cdot\sqrt{810091364106816}}\cr\approx \mathstrut & 4.15255158752 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 + 6*x^8 - 22*x^7 + 48*x^6 - 48*x^5 + 104*x^4 - 132*x^3 - 144*x^2 + 112*x + 144)
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^10 - 3*x^9 + 6*x^8 - 22*x^7 + 48*x^6 - 48*x^5 + 104*x^4 - 132*x^3 - 144*x^2 + 112*x + 144, 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^10 - 3*x^9 + 6*x^8 - 22*x^7 + 48*x^6 - 48*x^5 + 104*x^4 - 132*x^3 - 144*x^2 + 112*x + 144);
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^10 - 3*x^9 + 6*x^8 - 22*x^7 + 48*x^6 - 48*x^5 + 104*x^4 - 132*x^3 - 144*x^2 + 112*x + 144);
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
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 non-solvable group of order 60 |
| The 5 conjugacy class representatives for $A_{5}$ |
| Character table for $A_{5}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 5 sibling: | 5.1.4743684.1 |
| Degree 6 sibling: | 6.2.170772624.2 |
| Degree 12 sibling: | data not computed |
| Degree 15 sibling: | data not computed |
| Degree 20 sibling: | data not computed |
| Degree 30 sibling: | data not computed |
| Minimal sibling: | 5.1.4743684.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $[\ ]$ |
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ | |
| 3.6.7.1 | $x^{6} + 6 x^{2} + 6$ | $6$ | $1$ | $7$ | $S_3$ | $[3/2]_{2}$ | |
|
\(11\)
| 11.5.4.1 | $x^{5} + 55$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.5.4.1 | $x^{5} + 55$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |