Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^12 + 10*x^11 - 12*x^10 + 10*x^9 - 20*x^8 + 40*x^7 - 60*x^6 + 73*x^5 - 65*x^4 + 35*x^3 - 35*x^2 + 30*x - 9)
gp: K = bnfinit(y^15 - 5*y^12 + 10*y^11 - 12*y^10 + 10*y^9 - 20*y^8 + 40*y^7 - 60*y^6 + 73*y^5 - 65*y^4 + 35*y^3 - 35*y^2 + 30*y - 9, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 5*x^12 + 10*x^11 - 12*x^10 + 10*x^9 - 20*x^8 + 40*x^7 - 60*x^6 + 73*x^5 - 65*x^4 + 35*x^3 - 35*x^2 + 30*x - 9);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 5*x^12 + 10*x^11 - 12*x^10 + 10*x^9 - 20*x^8 + 40*x^7 - 60*x^6 + 73*x^5 - 65*x^4 + 35*x^3 - 35*x^2 + 30*x - 9)
\( x^{15} - 5 x^{12} + 10 x^{11} - 12 x^{10} + 10 x^{9} - 20 x^{8} + 40 x^{7} - 60 x^{6} + 73 x^{5} + \cdots - 9 \)
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: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-82170576019287109375\)
\(\medspace = -\,5^{16}\cdot 7^{5}\cdot 179^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.26\) | 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: | not computed | ||
| Ramified primes: |
\(5\), \(7\), \(179\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{63}a^{13}+\frac{1}{63}a^{12}+\frac{2}{21}a^{11}+\frac{3}{7}a^{10}+\frac{25}{63}a^{9}+\frac{22}{63}a^{8}+\frac{31}{63}a^{7}+\frac{16}{63}a^{6}-\frac{20}{63}a^{5}+\frac{1}{3}a^{4}+\frac{5}{21}a^{3}+\frac{13}{63}a^{2}-\frac{1}{21}a+\frac{3}{7}$, $\frac{1}{19161009}a^{14}+\frac{128510}{19161009}a^{13}-\frac{1969658}{19161009}a^{12}+\frac{855436}{6387003}a^{11}+\frac{3341185}{19161009}a^{10}-\frac{3366343}{19161009}a^{9}+\frac{8522282}{19161009}a^{8}-\frac{105619}{19161009}a^{7}-\frac{102325}{2737287}a^{6}+\frac{7373929}{19161009}a^{5}+\frac{1047254}{6387003}a^{4}-\frac{6560090}{19161009}a^{3}+\frac{8908117}{19161009}a^{2}+\frac{2544310}{6387003}a+\frac{520844}{2129001}$
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: | $7$ | 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: |
$\frac{1158865}{19161009}a^{14}+\frac{296342}{19161009}a^{13}+\frac{557104}{19161009}a^{12}-\frac{1932209}{6387003}a^{11}+\frac{10512202}{19161009}a^{10}-\frac{11914750}{19161009}a^{9}+\frac{15765203}{19161009}a^{8}-\frac{26530528}{19161009}a^{7}+\frac{6406331}{2737287}a^{6}-\frac{63807104}{19161009}a^{5}+\frac{27308678}{6387003}a^{4}-\frac{98266943}{19161009}a^{3}+\frac{53495191}{19161009}a^{2}-\frac{20753930}{6387003}a+\frac{2107982}{2129001}$, $\frac{83365}{709667}a^{14}+\frac{218395}{6387003}a^{13}+\frac{30934}{6387003}a^{12}-\frac{1325873}{2129001}a^{11}+\frac{686695}{709667}a^{10}-\frac{6837086}{6387003}a^{9}+\frac{6616354}{6387003}a^{8}-\frac{14288867}{6387003}a^{7}+\frac{3628342}{912429}a^{6}-\frac{36192578}{6387003}a^{5}+\frac{15535666}{2129001}a^{4}-\frac{12699605}{2129001}a^{3}+\frac{18950362}{6387003}a^{2}-\frac{7569965}{2129001}a+\frac{1177989}{709667}$, $\frac{88114}{709667}a^{14}+\frac{345868}{6387003}a^{13}+\frac{140041}{6387003}a^{12}-\frac{427006}{709667}a^{11}+\frac{732569}{709667}a^{10}-\frac{6470096}{6387003}a^{9}+\frac{4865368}{6387003}a^{8}-\frac{14155997}{6387003}a^{7}+\frac{3818755}{912429}a^{6}-\frac{37710083}{6387003}a^{5}+\frac{13263188}{2129001}a^{4}-\frac{10346887}{2129001}a^{3}+\frac{5678074}{6387003}a^{2}-\frac{1974603}{709667}a+\frac{1230032}{709667}$, $\frac{358544}{2129001}a^{14}+\frac{244274}{2129001}a^{13}+\frac{38900}{2129001}a^{12}-\frac{1749449}{2129001}a^{11}+\frac{2473811}{2129001}a^{10}-\frac{1943764}{2129001}a^{9}+\frac{889318}{2129001}a^{8}-\frac{1760952}{709667}a^{7}+\frac{1450474}{304143}a^{6}-\frac{12362660}{2129001}a^{5}+\frac{12582098}{2129001}a^{4}-\frac{8557373}{2129001}a^{3}+\frac{1587961}{2129001}a^{2}-\frac{5993636}{2129001}a+\frac{1113549}{709667}$, $\frac{3381124}{19161009}a^{14}+\frac{2256008}{19161009}a^{13}+\frac{1660333}{19161009}a^{12}-\frac{4823765}{6387003}a^{11}+\frac{24054868}{19161009}a^{10}-\frac{25178935}{19161009}a^{9}+\frac{13026365}{19161009}a^{8}-\frac{46445470}{19161009}a^{7}+\frac{12785456}{2737287}a^{6}-\frac{134769746}{19161009}a^{5}+\frac{47620346}{6387003}a^{4}-\frac{87892088}{19161009}a^{3}+\frac{831430}{19161009}a^{2}-\frac{11529110}{6387003}a+\frac{2124062}{2129001}$, $\frac{101965}{19161009}a^{14}-\frac{479005}{19161009}a^{13}-\frac{1126637}{19161009}a^{12}-\frac{639230}{6387003}a^{11}+\frac{1188505}{19161009}a^{10}-\frac{3106771}{19161009}a^{9}-\frac{693031}{19161009}a^{8}+\frac{1174724}{19161009}a^{7}+\frac{1273562}{2737287}a^{6}-\frac{5967680}{19161009}a^{5}+\frac{3208955}{6387003}a^{4}+\frac{11247340}{19161009}a^{3}+\frac{9808270}{19161009}a^{2}-\frac{976706}{6387003}a-\frac{71485}{2129001}$, $\frac{477955}{6387003}a^{14}+\frac{24091}{709667}a^{13}+\frac{58412}{6387003}a^{12}-\frac{738092}{2129001}a^{11}+\frac{3753304}{6387003}a^{10}-\frac{1520452}{2129001}a^{9}+\frac{245900}{2129001}a^{8}-\frac{2595749}{2129001}a^{7}+\frac{752909}{304143}a^{6}-\frac{23023624}{6387003}a^{5}+\frac{2394267}{709667}a^{4}-\frac{13370948}{6387003}a^{3}+\frac{9995939}{6387003}a^{2}-\frac{3261428}{2129001}a+\frac{570616}{709667}$
| 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: | \( 11224.8164015 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 11224.8164015 \cdot 1}{2\cdot\sqrt{82170576019287109375}}\cr\approx \mathstrut & 0.478718442637 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^12 + 10*x^11 - 12*x^10 + 10*x^9 - 20*x^8 + 40*x^7 - 60*x^6 + 73*x^5 - 65*x^4 + 35*x^3 - 35*x^2 + 30*x - 9)
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 - 5*x^12 + 10*x^11 - 12*x^10 + 10*x^9 - 20*x^8 + 40*x^7 - 60*x^6 + 73*x^5 - 65*x^4 + 35*x^3 - 35*x^2 + 30*x - 9, 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 - 5*x^12 + 10*x^11 - 12*x^10 + 10*x^9 - 20*x^8 + 40*x^7 - 60*x^6 + 73*x^5 - 65*x^4 + 35*x^3 - 35*x^2 + 30*x - 9);
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 - 5*x^12 + 10*x^11 - 12*x^10 + 10*x^9 - 20*x^8 + 40*x^7 - 60*x^6 + 73*x^5 - 65*x^4 + 35*x^3 - 35*x^2 + 30*x - 9);
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
$D_5\wr S_3$ (as 15T60):
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 6000 |
| The 40 conjugacy class representatives for $D_5\wr S_3$ |
| Character table for $D_5\wr S_3$ |
Intermediate fields
| 3.1.175.1 |
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 20 sibling: | 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 | $15$ | ${\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{5}$ | $15$ | $15$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{5}$ | $15$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.15.16.4 | $x^{15} + 15 x^{3} + 20 x^{2} + 5$ | $15$ | $1$ | $16$ | $((C_5^2 : C_3):C_2):C_2$ | $[7/6, 7/6]_{6}^{2}$ |
|
\(7\)
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.5.0.1 | $x^{5} + x + 4$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
|
\(179\)
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 179.2.1.1 | $x^{2} + 358$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 179.2.1.1 | $x^{2} + 358$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 179.5.0.1 | $x^{5} + 2 x + 177$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |