Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 + x^13 - 6*x^12 - 4*x^11 + 5*x^10 + 18*x^9 + 23*x^8 - 4*x^5 + 4*x^4 + 10*x^3 - 5*x^2 + 1)
gp: K = bnfinit(y^15 - y^14 + y^13 - 6*y^12 - 4*y^11 + 5*y^10 + 18*y^9 + 23*y^8 - 4*y^5 + 4*y^4 + 10*y^3 - 5*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - x^14 + x^13 - 6*x^12 - 4*x^11 + 5*x^10 + 18*x^9 + 23*x^8 - 4*x^5 + 4*x^4 + 10*x^3 - 5*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - x^14 + x^13 - 6*x^12 - 4*x^11 + 5*x^10 + 18*x^9 + 23*x^8 - 4*x^5 + 4*x^4 + 10*x^3 - 5*x^2 + 1)
\( x^{15} - x^{14} + x^{13} - 6x^{12} - 4x^{11} + 5x^{10} + 18x^{9} + 23x^{8} - 4x^{5} + 4x^{4} + 10x^{3} - 5x^{2} + 1 \)
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: |
\(-957206371560558183\)
\(\medspace = -\,3^{12}\cdot 23^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.80\) | 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: | $3^{7/6}23^{5/6}\approx 49.13768144738915$ | ||
| 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: | \(\Q(\sqrt{-23}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{23}a^{12}+\frac{10}{23}a^{11}-\frac{6}{23}a^{10}+\frac{5}{23}a^{9}-\frac{2}{23}a^{8}-\frac{11}{23}a^{7}-\frac{5}{23}a^{6}+\frac{3}{23}a^{5}+\frac{11}{23}a^{4}-\frac{2}{23}a^{3}-\frac{10}{23}a^{2}-\frac{1}{23}a+\frac{9}{23}$, $\frac{1}{23}a^{13}+\frac{9}{23}a^{11}-\frac{4}{23}a^{10}-\frac{6}{23}a^{9}+\frac{9}{23}a^{8}-\frac{10}{23}a^{7}+\frac{7}{23}a^{6}+\frac{4}{23}a^{5}+\frac{3}{23}a^{4}+\frac{10}{23}a^{3}+\frac{7}{23}a^{2}-\frac{4}{23}a+\frac{2}{23}$, $\frac{1}{89263}a^{14}-\frac{61}{3881}a^{13}-\frac{660}{89263}a^{12}-\frac{25531}{89263}a^{11}+\frac{7757}{89263}a^{10}+\frac{22539}{89263}a^{9}-\frac{558}{89263}a^{8}-\frac{28790}{89263}a^{7}-\frac{18225}{89263}a^{6}+\frac{18351}{89263}a^{5}-\frac{32005}{89263}a^{4}+\frac{33821}{89263}a^{3}-\frac{30022}{89263}a^{2}-\frac{21892}{89263}a-\frac{29412}{89263}$
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: |
$\frac{54744}{89263}a^{14}-\frac{78462}{89263}a^{13}+\frac{63166}{89263}a^{12}-\frac{357705}{89263}a^{11}-\frac{75789}{89263}a^{10}+\frac{438882}{89263}a^{9}+\frac{1063593}{89263}a^{8}+\frac{848160}{89263}a^{7}-\frac{859026}{89263}a^{6}-\frac{789192}{89263}a^{5}-\frac{834804}{89263}a^{4}+\frac{155037}{89263}a^{3}+\frac{537090}{89263}a^{2}-\frac{344376}{89263}a-\frac{54987}{89263}$, $\frac{23718}{89263}a^{14}-\frac{8422}{89263}a^{13}+\frac{29241}{89263}a^{12}-\frac{143187}{89263}a^{11}-\frac{165162}{89263}a^{10}-\frac{78201}{89263}a^{9}+\frac{410952}{89263}a^{8}+\frac{859026}{89263}a^{7}+\frac{789192}{89263}a^{6}+\frac{615828}{89263}a^{5}+\frac{63939}{89263}a^{4}+\frac{450}{3881}a^{3}+\frac{3072}{3881}a^{2}+\frac{54987}{89263}a+\frac{54744}{89263}$, $\frac{19351}{89263}a^{14}+\frac{21428}{89263}a^{13}+\frac{12354}{89263}a^{12}-\frac{96106}{89263}a^{11}-\frac{290805}{89263}a^{10}-\frac{277904}{89263}a^{9}+\frac{367779}{89263}a^{8}+\frac{1197608}{89263}a^{7}+\frac{1671087}{89263}a^{6}+\frac{1023285}{89263}a^{5}+\frac{474707}{89263}a^{4}-\frac{52717}{89263}a^{3}+\frac{107598}{89263}a^{2}+\frac{208037}{89263}a-\frac{34010}{89263}$, $\frac{19683}{89263}a^{14}-\frac{60149}{89263}a^{13}+\frac{88190}{89263}a^{12}-\frac{201081}{89263}a^{11}+\frac{227589}{89263}a^{10}+\frac{36837}{89263}a^{9}+\frac{8442}{3881}a^{8}-\frac{272668}{89263}a^{7}-\frac{397707}{89263}a^{6}+\frac{343472}{89263}a^{5}-\frac{40948}{89263}a^{4}+\frac{413842}{89263}a^{3}-\frac{33014}{89263}a^{2}-\frac{144165}{89263}a+\frac{140447}{89263}$, $\frac{20783}{89263}a^{14}-\frac{8358}{89263}a^{13}-\frac{5207}{89263}a^{12}-\frac{105240}{89263}a^{11}-\frac{166011}{89263}a^{10}+\frac{123291}{89263}a^{9}+\frac{531211}{89263}a^{8}+\frac{670375}{89263}a^{7}+\frac{54235}{89263}a^{6}-\frac{381256}{89263}a^{5}-\frac{294162}{89263}a^{4}-\frac{183998}{89263}a^{3}+\frac{78764}{89263}a^{2}+\frac{3718}{89263}a-\frac{120764}{89263}$, $\frac{24109}{89263}a^{14}-\frac{48584}{89263}a^{13}+\frac{50613}{89263}a^{12}-\frac{167162}{89263}a^{11}+\frac{50219}{89263}a^{10}+\frac{207991}{89263}a^{9}+\frac{282037}{89263}a^{8}+\frac{100241}{89263}a^{7}-\frac{515283}{89263}a^{6}+\frac{129975}{89263}a^{5}+\frac{4113}{89263}a^{4}+\frac{264059}{89263}a^{3}+\frac{165223}{89263}a^{2}-\frac{374090}{89263}a+\frac{120032}{89263}$, $\frac{13367}{89263}a^{14}-\frac{4790}{89263}a^{13}-\frac{707}{89263}a^{12}-\frac{51476}{89263}a^{11}-\frac{136693}{89263}a^{10}+\frac{93808}{89263}a^{9}+\frac{194546}{89263}a^{8}+\frac{465869}{89263}a^{7}+\frac{179202}{89263}a^{6}+\frac{150571}{89263}a^{5}+\frac{46129}{89263}a^{4}-\frac{229719}{89263}a^{3}+\frac{115518}{89263}a^{2}-\frac{115513}{89263}a+\frac{10620}{89263}$
| 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: | \( 627.503289611 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 627.503289611 \cdot 1}{2\cdot\sqrt{957206371560558183}}\cr\approx \mathstrut & 0.247954671576 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 + x^13 - 6*x^12 - 4*x^11 + 5*x^10 + 18*x^9 + 23*x^8 - 4*x^5 + 4*x^4 + 10*x^3 - 5*x^2 + 1)
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 - x^14 + x^13 - 6*x^12 - 4*x^11 + 5*x^10 + 18*x^9 + 23*x^8 - 4*x^5 + 4*x^4 + 10*x^3 - 5*x^2 + 1, 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 - x^14 + x^13 - 6*x^12 - 4*x^11 + 5*x^10 + 18*x^9 + 23*x^8 - 4*x^5 + 4*x^4 + 10*x^3 - 5*x^2 + 1);
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 - x^14 + x^13 - 6*x^12 - 4*x^11 + 5*x^10 + 18*x^9 + 23*x^8 - 4*x^5 + 4*x^4 + 10*x^3 - 5*x^2 + 1);
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
$S_3\times A_5$ (as 15T23):
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 360 |
| The 15 conjugacy class representatives for $A_5 \times S_3$ |
| Character table for $A_5 \times S_3$ |
Intermediate fields
| 3.1.23.1, 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 18 sibling: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Degree 45 sibling: | 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 | $15$ | R | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }$ | $15$ | ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.3.0.1}{3} }^{5}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.5.0.1}{5} }^{3}$ |
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 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.9.9.1 | $x^{9} + 90 x^{7} - 207 x^{6} + 540 x^{5} + 324 x^{4} + 243 x^{3} + 324 x^{2} + 162 x + 27$ | $3$ | $3$ | $9$ | $S_3\times C_3$ | $[3/2]_{2}^{3}$ | |
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.3.2.1 | $x^{3} + 23$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.6.5.2 | $x^{6} + 23$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ |