Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 - 8*x^8 + 78*x^6 + 44*x^5 - 124*x^4 - 323*x^3 + 747*x^2 + 1606*x + 2104)
gp: K = bnfinit(y^10 - y^9 - 8*y^8 + 78*y^6 + 44*y^5 - 124*y^4 - 323*y^3 + 747*y^2 + 1606*y + 2104, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - x^9 - 8*x^8 + 78*x^6 + 44*x^5 - 124*x^4 - 323*x^3 + 747*x^2 + 1606*x + 2104);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - x^9 - 8*x^8 + 78*x^6 + 44*x^5 - 124*x^4 - 323*x^3 + 747*x^2 + 1606*x + 2104)
\( x^{10} - x^{9} - 8x^{8} + 78x^{6} + 44x^{5} - 124x^{4} - 323x^{3} + 747x^{2} + 1606x + 2104 \)
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: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-24871763102042647\)
\(\medspace = -\,7^{5}\cdot 1216489^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(43.61\) | 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: | $7^{1/2}1216489^{1/2}\approx 2918.1197713596334$ | ||
| Ramified primes: |
\(7\), \(1216489\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{16}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}-\frac{2}{5}a^{5}+\frac{1}{5}a^{3}+\frac{1}{5}a^{2}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{10}a^{8}-\frac{1}{5}a^{6}-\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}+\frac{3}{10}a$, $\frac{1}{5054726020}a^{9}+\frac{37232741}{2527363010}a^{8}-\frac{79491047}{2527363010}a^{7}+\frac{1234998513}{2527363010}a^{6}+\frac{108728108}{252736301}a^{5}+\frac{350963977}{1263681505}a^{4}-\frac{109128538}{1263681505}a^{3}+\frac{1415990317}{5054726020}a^{2}+\frac{226874923}{505472602}a-\frac{517267384}{1263681505}$
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
$C_{120}$, which has order $120$
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: | $4$ | 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: |
$\frac{14731303}{2527363010}a^{9}-\frac{14820871}{1263681505}a^{8}-\frac{57412727}{1263681505}a^{7}+\frac{96937177}{1263681505}a^{6}+\frac{505039222}{1263681505}a^{5}-\frac{386868087}{1263681505}a^{4}-\frac{991648428}{1263681505}a^{3}+\frac{11294015}{505472602}a^{2}+\frac{882921458}{252736301}a+\frac{6135271603}{1263681505}$, $\frac{2408648}{1263681505}a^{9}-\frac{5596441}{1263681505}a^{8}-\frac{12706568}{1263681505}a^{7}+\frac{11435282}{1263681505}a^{6}+\frac{181211092}{1263681505}a^{5}-\frac{134541358}{1263681505}a^{4}-\frac{147923794}{1263681505}a^{3}-\frac{304101471}{1263681505}a^{2}+\frac{1004071026}{1263681505}a+\frac{668928589}{1263681505}$, $\frac{9914007}{2527363010}a^{9}+\frac{1844886}{252736301}a^{8}+\frac{44706159}{1263681505}a^{7}-\frac{17100379}{252736301}a^{6}-\frac{64765626}{252736301}a^{5}+\frac{252326729}{1263681505}a^{4}+\frac{843724634}{1263681505}a^{3}-\frac{664673017}{2527363010}a^{2}-\frac{3410536264}{1263681505}a-\frac{4202661509}{1263681505}$, $\frac{41912947}{2527363010}a^{9}-\frac{38087027}{1263681505}a^{8}-\frac{169863979}{1263681505}a^{7}+\frac{270315169}{1263681505}a^{6}+\frac{283798138}{252736301}a^{5}-\frac{825868661}{1263681505}a^{4}-\frac{2800287156}{1263681505}a^{3}-\frac{128018025}{505472602}a^{2}+\frac{11050147438}{1263681505}a+\frac{28361337119}{1263681505}$
| 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: | \( 1103.1135300938238 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 1103.1135300938238 \cdot 120}{2\cdot\sqrt{24871763102042647}}\cr\approx \mathstrut & 4.10976999731846 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 - 8*x^8 + 78*x^6 + 44*x^5 - 124*x^4 - 323*x^3 + 747*x^2 + 1606*x + 2104)
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 - x^9 - 8*x^8 + 78*x^6 + 44*x^5 - 124*x^4 - 323*x^3 + 747*x^2 + 1606*x + 2104, 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 - x^9 - 8*x^8 + 78*x^6 + 44*x^5 - 124*x^4 - 323*x^3 + 747*x^2 + 1606*x + 2104);
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 - x^9 - 8*x^8 + 78*x^6 + 44*x^5 - 124*x^4 - 323*x^3 + 747*x^2 + 1606*x + 2104);
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
$C_2\times S_5$ (as 10T22):
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 240 |
| The 14 conjugacy class representatives for $S_5\times C_2$ |
| Character table for $S_5\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), 5.5.1216489.1 |
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 10 sibling: | data not computed |
| Degree 12 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| 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 | ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }^{4}$ | ${\href{/padicField/3.10.0.1}{10} }$ | ${\href{/padicField/5.2.0.1}{2} }^{5}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.10.5.2 | $x^{10} + 35 x^{8} + 492 x^{6} + 8 x^{5} + 3360 x^{4} - 560 x^{3} + 11516 x^{2} + 1968 x + 17516$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
|
\(1216489\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |