Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 4*x^13 - 10*x^12 + 22*x^11 - 33*x^10 + 40*x^9 - 63*x^8 + 81*x^7 - 81*x^6 + 63*x^5 - 69*x^4 + 57*x^3 - 24*x^2 + 3)
gp: K = bnfinit(y^15 - 2*y^14 + 4*y^13 - 10*y^12 + 22*y^11 - 33*y^10 + 40*y^9 - 63*y^8 + 81*y^7 - 81*y^6 + 63*y^5 - 69*y^4 + 57*y^3 - 24*y^2 + 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 2*x^14 + 4*x^13 - 10*x^12 + 22*x^11 - 33*x^10 + 40*x^9 - 63*x^8 + 81*x^7 - 81*x^6 + 63*x^5 - 69*x^4 + 57*x^3 - 24*x^2 + 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 2*x^14 + 4*x^13 - 10*x^12 + 22*x^11 - 33*x^10 + 40*x^9 - 63*x^8 + 81*x^7 - 81*x^6 + 63*x^5 - 69*x^4 + 57*x^3 - 24*x^2 + 3)
\( x^{15} - 2 x^{14} + 4 x^{13} - 10 x^{12} + 22 x^{11} - 33 x^{10} + 40 x^{9} - 63 x^{8} + 81 x^{7} + \cdots + 3 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[3, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(57352136505929721\)
\(\medspace = 3^{18}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.10\) | 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: | $3^{97/54}23^{2/3}\approx 58.192859749017636$ | ||
| Ramified primes: |
\(3\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{23465906}a^{14}+\frac{2240267}{23465906}a^{13}-\frac{2566391}{11732953}a^{12}-\frac{1104218}{11732953}a^{11}+\frac{2247013}{23465906}a^{10}-\frac{4321609}{23465906}a^{9}-\frac{3175301}{23465906}a^{8}-\frac{1626737}{11732953}a^{7}-\frac{5745389}{23465906}a^{6}-\frac{1583690}{11732953}a^{5}+\frac{1692465}{23465906}a^{4}+\frac{1356674}{11732953}a^{3}+\frac{4693738}{11732953}a^{2}-\frac{1004432}{11732953}a-\frac{5214056}{11732953}$
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: | $8$ | 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: |
$\frac{2121397}{11732953}a^{14}-\frac{2901785}{23465906}a^{13}+\frac{4831572}{11732953}a^{12}-\frac{13105145}{11732953}a^{11}+\frac{24622992}{11732953}a^{10}-\frac{23950351}{11732953}a^{9}+\frac{25494957}{11732953}a^{8}-\frac{129824339}{23465906}a^{7}+\frac{80636135}{23465906}a^{6}-\frac{50612781}{23465906}a^{5}+\frac{959028}{11732953}a^{4}-\frac{51502832}{11732953}a^{3}-\frac{24094941}{11732953}a^{2}+\frac{55052445}{23465906}a-\frac{34040249}{23465906}$, $\frac{1377556}{11732953}a^{14}-\frac{1914232}{11732953}a^{13}+\frac{5223316}{11732953}a^{12}-\frac{25491045}{23465906}a^{11}+\frac{55824301}{23465906}a^{10}-\frac{40186075}{11732953}a^{9}+\frac{54195386}{11732953}a^{8}-\frac{190218161}{23465906}a^{7}+\frac{217764805}{23465906}a^{6}-\frac{231915293}{23465906}a^{5}+\frac{92355581}{11732953}a^{4}-\frac{249363269}{23465906}a^{3}+\frac{76813599}{11732953}a^{2}-\frac{58854279}{23465906}a+\frac{3148996}{11732953}$, $\frac{329141}{11732953}a^{14}+\frac{845771}{23465906}a^{13}+\frac{2605557}{23465906}a^{12}-\frac{2125487}{23465906}a^{11}+\frac{6559909}{23465906}a^{10}-\frac{6966593}{23465906}a^{9}+\frac{17282927}{23465906}a^{8}-\frac{20264383}{11732953}a^{7}+\frac{19504899}{23465906}a^{6}-\frac{30013577}{11732953}a^{5}+\frac{14212984}{11732953}a^{4}-\frac{38308292}{11732953}a^{3}+\frac{463284}{11732953}a^{2}-\frac{36543783}{23465906}a+\frac{3659969}{11732953}$, $\frac{1280919}{23465906}a^{14}+\frac{1852445}{23465906}a^{13}-\frac{221789}{11732953}a^{12}-\frac{931431}{23465906}a^{11}-\frac{2127171}{11732953}a^{10}+\frac{23078541}{23465906}a^{9}-\frac{28292357}{23465906}a^{8}+\frac{2646417}{23465906}a^{7}-\frac{29578768}{11732953}a^{6}+\frac{62726321}{23465906}a^{5}-\frac{57548193}{23465906}a^{4}+\frac{8470293}{23465906}a^{3}-\frac{42386474}{11732953}a^{2}+\frac{43987085}{23465906}a+\frac{637585}{11732953}$, $\frac{450695}{11732953}a^{14}-\frac{2134850}{11732953}a^{13}+\frac{5256557}{23465906}a^{12}-\frac{14121201}{23465906}a^{11}+\frac{34036445}{23465906}a^{10}-\frac{56075745}{23465906}a^{9}+\frac{57583007}{23465906}a^{8}-\frac{79457181}{23465906}a^{7}+\frac{68097651}{11732953}a^{6}-\frac{100403569}{23465906}a^{5}+\frac{37971598}{11732953}a^{4}-\frac{33314330}{11732953}a^{3}+\frac{56041738}{11732953}a^{2}+\frac{11823671}{11732953}a-\frac{22710401}{23465906}$, $\frac{232143}{11732953}a^{14}-\frac{839544}{11732953}a^{13}+\frac{630089}{11732953}a^{12}-\frac{1577013}{11732953}a^{11}+\frac{4714385}{11732953}a^{10}-\frac{5131822}{11732953}a^{9}-\frac{1127818}{11732953}a^{8}+\frac{2435734}{11732953}a^{7}+\frac{3326601}{11732953}a^{6}+\frac{7336217}{11732953}a^{5}-\frac{19227569}{11732953}a^{4}+\frac{12895912}{11732953}a^{3}-\frac{6650293}{11732953}a^{2}+\frac{17700292}{11732953}a+\frac{56662}{11732953}$, $\frac{2121397}{11732953}a^{14}-\frac{2901785}{23465906}a^{13}+\frac{4831572}{11732953}a^{12}-\frac{13105145}{11732953}a^{11}+\frac{24622992}{11732953}a^{10}-\frac{23950351}{11732953}a^{9}+\frac{25494957}{11732953}a^{8}-\frac{129824339}{23465906}a^{7}+\frac{80636135}{23465906}a^{6}-\frac{50612781}{23465906}a^{5}+\frac{959028}{11732953}a^{4}-\frac{51502832}{11732953}a^{3}-\frac{24094941}{11732953}a^{2}+\frac{78518351}{23465906}a-\frac{34040249}{23465906}$, $\frac{3933711}{11732953}a^{14}-\frac{10518349}{23465906}a^{13}+\frac{23363169}{23465906}a^{12}-\frac{60118519}{23465906}a^{11}+\frac{124026527}{23465906}a^{10}-\frac{158785645}{23465906}a^{9}+\frac{171123377}{23465906}a^{8}-\frac{155988674}{11732953}a^{7}+\frac{329598227}{23465906}a^{6}-\frac{147029232}{11732953}a^{5}+\frac{85598637}{11732953}a^{4}-\frac{139929873}{11732953}a^{3}+\frac{72240228}{11732953}a^{2}-\frac{23279689}{23465906}a-\frac{17951677}{11732953}$
| 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: | \( 279.393832589 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 279.393832589 \cdot 1}{2\cdot\sqrt{57352136505929721}}\cr\approx \mathstrut & 0.287131748826 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 4*x^13 - 10*x^12 + 22*x^11 - 33*x^10 + 40*x^9 - 63*x^8 + 81*x^7 - 81*x^6 + 63*x^5 - 69*x^4 + 57*x^3 - 24*x^2 + 3)
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^15 - 2*x^14 + 4*x^13 - 10*x^12 + 22*x^11 - 33*x^10 + 40*x^9 - 63*x^8 + 81*x^7 - 81*x^6 + 63*x^5 - 69*x^4 + 57*x^3 - 24*x^2 + 3, 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^15 - 2*x^14 + 4*x^13 - 10*x^12 + 22*x^11 - 33*x^10 + 40*x^9 - 63*x^8 + 81*x^7 - 81*x^6 + 63*x^5 - 69*x^4 + 57*x^3 - 24*x^2 + 3);
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^15 - 2*x^14 + 4*x^13 - 10*x^12 + 22*x^11 - 33*x^10 + 40*x^9 - 63*x^8 + 81*x^7 - 81*x^6 + 63*x^5 - 69*x^4 + 57*x^3 - 24*x^2 + 3);
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
$C_3^4:A_5$ (as 15T53):
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 4860 |
| The 24 conjugacy class representatives for $C_3^4:A_5$ |
| Character table for $C_3^4:A_5$ |
Intermediate fields
| 5.1.42849.1 |
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 15 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 45 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 | ${\href{/padicField/2.5.0.1}{5} }^{3}$ | R | ${\href{/padicField/5.5.0.1}{5} }^{3}$ | ${\href{/padicField/7.5.0.1}{5} }^{3}$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{5}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.5.0.1}{5} }^{3}$ | ${\href{/padicField/41.5.0.1}{5} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{3}$ | ${\href{/padicField/59.5.0.1}{5} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.9.15.23 | $x^{9} + 3 x^{8} + 3 x^{7} + 6 x^{3} + 3$ | $9$ | $1$ | $15$ | $C_3 \wr S_3 $ | $[3/2, 3/2, 2]_{2}^{3}$ | |
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.9.6.1 | $x^{9} + 6 x^{7} + 123 x^{6} + 12 x^{5} + 78 x^{4} - 6127 x^{3} + 492 x^{2} - 6888 x + 69105$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |