Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 15*x^13 + 90*x^11 - 2*x^10 - 275*x^9 + 20*x^8 + 450*x^7 - 70*x^6 - 375*x^5 + 100*x^4 + 125*x^3 - 50*x^2 + 8)
gp: K = bnfinit(y^15 - 15*y^13 + 90*y^11 - 2*y^10 - 275*y^9 + 20*y^8 + 450*y^7 - 70*y^6 - 375*y^5 + 100*y^4 + 125*y^3 - 50*y^2 + 8, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 15*x^13 + 90*x^11 - 2*x^10 - 275*x^9 + 20*x^8 + 450*x^7 - 70*x^6 - 375*x^5 + 100*x^4 + 125*x^3 - 50*x^2 + 8);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 15*x^13 + 90*x^11 - 2*x^10 - 275*x^9 + 20*x^8 + 450*x^7 - 70*x^6 - 375*x^5 + 100*x^4 + 125*x^3 - 50*x^2 + 8)
\( x^{15} - 15 x^{13} + 90 x^{11} - 2 x^{10} - 275 x^{9} + 20 x^{8} + 450 x^{7} - 70 x^{6} - 375 x^{5} + \cdots + 8 \)
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: | $[5, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-804542875000000000000\)
\(\medspace = -\,2^{12}\cdot 5^{15}\cdot 23^{5}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(24.76\) | 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: |
\(2\), \(5\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-115}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{7}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{2}$
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 $1$ (assuming GRH)
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: | $9$ | 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: |
$\frac{1}{2}a^{5}+\frac{5}{2}a^{3}-\frac{5}{2}a$, $\frac{1}{4}a^{13}-3a^{11}+\frac{55}{4}a^{9}-\frac{1}{2}a^{8}-30a^{7}+\frac{7}{2}a^{6}+\frac{125}{4}a^{5}-\frac{15}{2}a^{4}-\frac{25}{2}a^{3}+5a^{2}-1$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{12}+\frac{13}{4}a^{11}+\frac{13}{4}a^{10}-\frac{65}{4}a^{9}-\frac{63}{4}a^{8}+\frac{159}{4}a^{7}+\frac{141}{4}a^{6}-\frac{205}{4}a^{5}-\frac{149}{4}a^{4}+\frac{141}{4}a^{3}+\frac{71}{4}a^{2}-\frac{19}{2}a-3$, $\frac{1}{4}a^{11}+3a^{9}-\frac{55}{4}a^{7}+\frac{1}{2}a^{6}+30a^{5}-\frac{7}{2}a^{4}-\frac{125}{4}a^{3}+\frac{15}{2}a^{2}+\frac{25}{2}a-5$, $\frac{1}{4}a^{11}-3a^{9}+\frac{55}{4}a^{7}-\frac{1}{2}a^{6}-30a^{5}+\frac{7}{2}a^{4}+\frac{125}{4}a^{3}-\frac{15}{2}a^{2}-\frac{25}{2}a+4$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{7}{2}a^{12}+\frac{7}{2}a^{11}+19a^{10}-\frac{77}{4}a^{9}-\frac{99}{2}a^{8}+\frac{105}{2}a^{7}+60a^{6}-\frac{289}{4}a^{5}-\frac{43}{2}a^{4}+\frac{83}{2}a^{3}-\frac{41}{4}a^{2}-a+2$, $\frac{1}{4}a^{14}+\frac{15}{4}a^{12}+\frac{1}{4}a^{11}-\frac{45}{2}a^{10}-\frac{5}{2}a^{9}+\frac{275}{4}a^{8}+\frac{35}{4}a^{7}-113a^{6}-13a^{5}+\frac{389}{4}a^{4}+\frac{39}{4}a^{3}-\frac{151}{4}a^{2}-\frac{13}{2}a+2$, $\frac{1}{4}a^{12}-\frac{13}{4}a^{10}+\frac{1}{2}a^{9}+\frac{63}{4}a^{8}-\frac{9}{2}a^{7}-\frac{141}{4}a^{6}+\frac{29}{2}a^{5}+\frac{145}{4}a^{4}-20a^{3}-\frac{51}{4}a^{2}+9a-1$, $\frac{1}{2}a^{7}+\frac{1}{2}a^{6}-3a^{5}-\frac{5}{2}a^{4}+5a^{3}+\frac{5}{2}a^{2}-\frac{5}{2}a+1$
| 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: | \( 59566.61941447335 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 59566.61941447335 \cdot 1}{2\cdot\sqrt{804542875000000000000}}\cr\approx \mathstrut & 0.329039230745383 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 15*x^13 + 90*x^11 - 2*x^10 - 275*x^9 + 20*x^8 + 450*x^7 - 70*x^6 - 375*x^5 + 100*x^4 + 125*x^3 - 50*x^2 + 8)
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 - 15*x^13 + 90*x^11 - 2*x^10 - 275*x^9 + 20*x^8 + 450*x^7 - 70*x^6 - 375*x^5 + 100*x^4 + 125*x^3 - 50*x^2 + 8, 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 - 15*x^13 + 90*x^11 - 2*x^10 - 275*x^9 + 20*x^8 + 450*x^7 - 70*x^6 - 375*x^5 + 100*x^4 + 125*x^3 - 50*x^2 + 8);
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 - 15*x^13 + 90*x^11 - 2*x^10 - 275*x^9 + 20*x^8 + 450*x^7 - 70*x^6 - 375*x^5 + 100*x^4 + 125*x^3 - 50*x^2 + 8);
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
$D_5^3.D_6$ (as 15T68):
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 12000 |
| The 38 conjugacy class representatives for $D_5^3.D_6$ |
| Character table for $D_5^3.D_6$ |
Intermediate fields
| 3.1.23.1 |
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 | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | R | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.12.12.12 | $x^{12} + 84 x^{10} - 88 x^{9} + 1660 x^{8} - 1568 x^{7} + 13920 x^{6} - 4928 x^{5} + 47280 x^{4} + 23936 x^{3} + 63552 x^{2} + 32896 x + 51520$ | $2$ | $6$ | $12$ | 12T29 | $[2, 2, 2]^{6}$ | |
|
\(5\)
| 5.5.5.1 | $x^{5} + 20 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $[5/4]_{4}$ |
| 5.10.10.14 | $x^{10} + 10 x^{6} + 10 x^{5} - 25 x^{2} + 50 x + 25$ | $5$ | $2$ | $10$ | $(C_5^2 : C_4) : C_2$ | $[5/4, 5/4]_{4}^{2}$ | |
|
\(23\)
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.8.4.1 | $x^{8} + 98 x^{6} + 38 x^{5} + 3331 x^{4} - 1634 x^{3} + 44919 x^{2} - 57494 x + 224528$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |