Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 25*x^13 - 27*x^12 + 183*x^11 + 358*x^10 - 255*x^9 - 1041*x^8 - 456*x^7 + 700*x^6 + 710*x^5 + 78*x^4 - 157*x^3 - 80*x^2 - 15*x - 1)
gp: K = bnfinit(y^15 - 25*y^13 - 27*y^12 + 183*y^11 + 358*y^10 - 255*y^9 - 1041*y^8 - 456*y^7 + 700*y^6 + 710*y^5 + 78*y^4 - 157*y^3 - 80*y^2 - 15*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 25*x^13 - 27*x^12 + 183*x^11 + 358*x^10 - 255*x^9 - 1041*x^8 - 456*x^7 + 700*x^6 + 710*x^5 + 78*x^4 - 157*x^3 - 80*x^2 - 15*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 25*x^13 - 27*x^12 + 183*x^11 + 358*x^10 - 255*x^9 - 1041*x^8 - 456*x^7 + 700*x^6 + 710*x^5 + 78*x^4 - 157*x^3 - 80*x^2 - 15*x - 1)
\( x^{15} - 25 x^{13} - 27 x^{12} + 183 x^{11} + 358 x^{10} - 255 x^{9} - 1041 x^{8} - 456 x^{7} + 700 x^{6} + \cdots - 1 \)
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: | $[15, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(127667878299171450781696\)
\(\medspace = 2^{18}\cdot 887^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(34.71\) | 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: | $2^{3/2}887^{1/2}\approx 84.23775875461075$ | ||
| Ramified primes: |
\(2\), \(887\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{57}a^{13}-\frac{16}{57}a^{12}-\frac{4}{57}a^{11}-\frac{22}{57}a^{10}-\frac{7}{57}a^{9}-\frac{1}{19}a^{8}+\frac{13}{57}a^{7}+\frac{26}{57}a^{6}+\frac{2}{19}a^{5}+\frac{23}{57}a^{4}+\frac{5}{19}a^{3}+\frac{1}{3}a^{2}+\frac{4}{57}a+\frac{8}{57}$, $\frac{1}{3249}a^{14}-\frac{8}{3249}a^{13}+\frac{13}{1083}a^{12}-\frac{113}{1083}a^{11}-\frac{118}{1083}a^{10}-\frac{59}{3249}a^{9}+\frac{217}{3249}a^{8}+\frac{472}{3249}a^{7}-\frac{983}{3249}a^{6}-\frac{1183}{3249}a^{5}+\frac{427}{3249}a^{4}-\frac{89}{3249}a^{3}+\frac{185}{1083}a^{2}-\frac{1271}{3249}a+\frac{406}{3249}$
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: | $14$ | 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: |
$a$, $a^{14}-a^{13}-24a^{12}-3a^{11}+186a^{10}+172a^{9}-427a^{8}-614a^{7}+158a^{6}+542a^{5}+168a^{4}-90a^{3}-67a^{2}-13a-1$, $\frac{7}{361}a^{14}+\frac{1067}{1083}a^{13}-\frac{1613}{1083}a^{12}-\frac{26138}{1083}a^{11}+\frac{1135}{1083}a^{10}+\frac{205633}{1083}a^{9}+\frac{58044}{361}a^{8}-\frac{477628}{1083}a^{7}-\frac{652412}{1083}a^{6}+\frac{62057}{361}a^{5}+\frac{577789}{1083}a^{4}+\frac{60424}{361}a^{3}-\frac{90508}{1083}a^{2}-\frac{80233}{1083}a-\frac{14084}{1083}$, $\frac{2150}{1083}a^{14}-\frac{1088}{1083}a^{13}-\frac{52646}{1083}a^{12}-\frac{31949}{1083}a^{11}+\frac{396304}{1083}a^{10}+\frac{189067}{361}a^{9}-\frac{731932}{1083}a^{8}-\frac{1770239}{1083}a^{7}-\frac{107459}{361}a^{6}+\frac{1320979}{1083}a^{5}+\frac{316188}{361}a^{4}-\frac{2737}{1083}a^{3}-\frac{249667}{1083}a^{2}-\frac{100409}{1083}a-\frac{3603}{361}$, $\frac{5625}{361}a^{14}-\frac{12545}{1083}a^{13}-\frac{414178}{1083}a^{12}-\frac{146728}{1083}a^{11}+\frac{3237659}{1083}a^{10}+\frac{3653162}{1083}a^{9}-\frac{2445406}{361}a^{8}-\frac{12500546}{1083}a^{7}+\frac{2308952}{1083}a^{6}+\frac{3797810}{361}a^{5}+\frac{3319403}{1083}a^{4}-\frac{776049}{361}a^{3}-\frac{1275572}{1083}a^{2}-\frac{160373}{1083}a-\frac{2449}{1083}$, $\frac{52274}{1083}a^{14}-\frac{22175}{1083}a^{13}-\frac{1295510}{1083}a^{12}-\frac{864713}{1083}a^{11}+\frac{9888376}{1083}a^{10}+\frac{4844813}{361}a^{9}-\frac{19151656}{1083}a^{8}-\frac{46118411}{1083}a^{7}-\frac{1707851}{361}a^{6}+\frac{37978885}{1083}a^{5}+\frac{7181977}{361}a^{4}-\frac{4307992}{1083}a^{3}-\frac{6319000}{1083}a^{2}-\frac{1666385}{1083}a-\frac{45829}{361}$, $\frac{1067}{1083}a^{14}-\frac{5}{1083}a^{13}-\frac{26654}{1083}a^{12}-\frac{28700}{1083}a^{11}+\frac{194866}{1083}a^{10}+\frac{126975}{361}a^{9}-\frac{269491}{1083}a^{8}-\frac{1105277}{1083}a^{7}-\frac{164497}{361}a^{6}+\frac{733993}{1083}a^{5}+\frac{255540}{361}a^{4}+\frac{94733}{1083}a^{3}-\frac{176023}{1083}a^{2}-\frac{88496}{1083}a-\frac{4325}{361}$, $\frac{3254}{1083}a^{14}-\frac{729}{361}a^{13}-\frac{80251}{1083}a^{12}-\frac{33178}{1083}a^{11}+\frac{625952}{1083}a^{10}+\frac{737095}{1083}a^{9}-\frac{1387378}{1083}a^{8}-\frac{818601}{361}a^{7}+\frac{329143}{1083}a^{6}+\frac{2120128}{1083}a^{5}+\frac{753091}{1083}a^{4}-\frac{319645}{1083}a^{3}-\frac{278444}{1083}a^{2}-\frac{20143}{361}a-\frac{2150}{1083}$, $\frac{3964}{361}a^{14}-\frac{2150}{1083}a^{13}-\frac{296212}{1083}a^{12}-\frac{268438}{1083}a^{11}+\frac{2208185}{1083}a^{10}+\frac{3861032}{1083}a^{9}-\frac{1199887}{361}a^{8}-\frac{11647640}{1083}a^{7}-\frac{3652513}{1083}a^{6}+\frac{2882259}{361}a^{5}+\frac{7122341}{1083}a^{4}-\frac{6996}{361}a^{3}-\frac{1864307}{1083}a^{2}-\frac{701693}{1083}a-\frac{76888}{1083}$, $\frac{3422}{361}a^{14}+\frac{484}{1083}a^{13}-\frac{260788}{1083}a^{12}-\frac{283291}{1083}a^{11}+\frac{1961507}{1083}a^{10}+\frac{3735008}{1083}a^{9}-\frac{1062008}{361}a^{8}-\frac{11209220}{1083}a^{7}-\frac{3351646}{1083}a^{6}+\frac{2905964}{361}a^{5}+\frac{6472100}{1083}a^{4}-\frac{178854}{361}a^{3}-\frac{1734638}{1083}a^{2}-\frac{532805}{1083}a-\frac{50023}{1083}$, $\frac{1067}{1083}a^{14}-\frac{1088}{1083}a^{13}-\frac{25571}{1083}a^{12}-\frac{2708}{1083}a^{11}+\frac{198115}{1083}a^{10}+\frac{59829}{361}a^{9}-\frac{455767}{1083}a^{8}-\frac{642836}{1083}a^{7}+\frac{57157}{361}a^{6}+\frac{562879}{1083}a^{5}+\frac{59878}{361}a^{4}-\frac{87211}{1083}a^{3}-\frac{78553}{1083}a^{2}-\frac{14852}{1083}a+\frac{7}{361}$, $\frac{1067}{1083}a^{14}-\frac{5}{1083}a^{13}-\frac{26654}{1083}a^{12}-\frac{28700}{1083}a^{11}+\frac{194866}{1083}a^{10}+\frac{126975}{361}a^{9}-\frac{269491}{1083}a^{8}-\frac{1105277}{1083}a^{7}-\frac{164497}{361}a^{6}+\frac{733993}{1083}a^{5}+\frac{255540}{361}a^{4}+\frac{94733}{1083}a^{3}-\frac{176023}{1083}a^{2}-\frac{88496}{1083}a-\frac{3964}{361}$, $\frac{3964}{361}a^{14}-\frac{2150}{1083}a^{13}-\frac{296212}{1083}a^{12}-\frac{268438}{1083}a^{11}+\frac{2208185}{1083}a^{10}+\frac{3861032}{1083}a^{9}-\frac{1199887}{361}a^{8}-\frac{11647640}{1083}a^{7}-\frac{3652513}{1083}a^{6}+\frac{2882259}{361}a^{5}+\frac{7122341}{1083}a^{4}-\frac{6996}{361}a^{3}-\frac{1864307}{1083}a^{2}-\frac{701693}{1083}a-\frac{77971}{1083}$, $\frac{89881}{3249}a^{14}-\frac{24446}{3249}a^{13}-\frac{745892}{1083}a^{12}-\frac{607913}{1083}a^{11}+\frac{5628896}{1083}a^{10}+\frac{27633673}{3249}a^{9}-\frac{30011780}{3249}a^{8}-\frac{85345487}{3249}a^{7}-\frac{18803219}{3249}a^{6}+\frac{67371209}{3249}a^{5}+\frac{46164304}{3249}a^{4}-\frac{4828898}{3249}a^{3}-\frac{4251172}{1083}a^{2}-\frac{3885953}{3249}a-\frac{357446}{3249}$
| 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: | \( 5340385.5843 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 5340385.5843 \cdot 1}{2\cdot\sqrt{127667878299171450781696}}\cr\approx \mathstrut & 0.24487910750 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 25*x^13 - 27*x^12 + 183*x^11 + 358*x^10 - 255*x^9 - 1041*x^8 - 456*x^7 + 700*x^6 + 710*x^5 + 78*x^4 - 157*x^3 - 80*x^2 - 15*x - 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 - 25*x^13 - 27*x^12 + 183*x^11 + 358*x^10 - 255*x^9 - 1041*x^8 - 456*x^7 + 700*x^6 + 710*x^5 + 78*x^4 - 157*x^3 - 80*x^2 - 15*x - 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 - 25*x^13 - 27*x^12 + 183*x^11 + 358*x^10 - 255*x^9 - 1041*x^8 - 456*x^7 + 700*x^6 + 710*x^5 + 78*x^4 - 157*x^3 - 80*x^2 - 15*x - 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 - 25*x^13 - 27*x^12 + 183*x^11 + 358*x^10 - 255*x^9 - 1041*x^8 - 456*x^7 + 700*x^6 + 710*x^5 + 78*x^4 - 157*x^3 - 80*x^2 - 15*x - 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
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 60 |
| The 5 conjugacy class representatives for $A_5$ |
| Character table for $A_5$ |
Intermediate fields
| 5.5.50353216.1 |
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 5 sibling: | 5.5.50353216.1 |
| Degree 6 sibling: | 6.6.50353216.1 |
| Degree 10 sibling: | 10.10.2535446361542656.1 |
| Degree 12 sibling: | deg 12 |
| Degree 20 sibling: | deg 20 |
| Degree 30 sibling: | data not computed |
| Minimal sibling: | 5.5.50353216.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{5}$ | ${\href{/padicField/5.5.0.1}{5} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{5}$ | ${\href{/padicField/11.5.0.1}{5} }^{3}$ | ${\href{/padicField/13.5.0.1}{5} }^{3}$ | ${\href{/padicField/17.5.0.1}{5} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{5}$ | ${\href{/padicField/29.5.0.1}{5} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.5.0.1}{5} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{5}$ | ${\href{/padicField/43.5.0.1}{5} }^{3}$ | ${\href{/padicField/47.5.0.1}{5} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{5}$ | ${\href{/padicField/59.3.0.1}{3} }^{5}$ |
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 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.12.18.59 | $x^{12} + 6 x^{11} + 22 x^{10} + 56 x^{9} + 126 x^{8} + 240 x^{7} + 332 x^{6} - 18 x^{5} - 459 x^{4} - 394 x^{3} - 344 x^{2} + 138 x + 423$ | $4$ | $3$ | $18$ | $A_4$ | $[2, 2]^{3}$ | |
|
\(887\)
| $\Q_{887}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{887}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{887}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |