Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 145860*x^10 - 243079056*x^5 - 4696791102432)
gp: K = bnfinit(y^15 - 145860*y^10 - 243079056*y^5 - 4696791102432, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 145860*x^10 - 243079056*x^5 - 4696791102432);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 145860*x^10 - 243079056*x^5 - 4696791102432)
\( x^{15} - 145860x^{10} - 243079056x^{5} - 4696791102432 \)
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: |
\(-5617330134094061678662632074276610010758000000000\)
\(\medspace = -\,2^{10}\cdot 3^{12}\cdot 5^{9}\cdot 11^{13}\cdot 10889^{5}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(1778.15\) | 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: | $2^{2/3}3^{4/5}5^{3/4}11^{9/10}10889^{1/2}\approx 11544.060292982782$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(11\), \(10889\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-598895}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{6}a^{5}$, $\frac{1}{12}a^{6}$, $\frac{1}{12}a^{7}$, $\frac{1}{12}a^{8}$, $\frac{1}{24}a^{9}$, $\frac{1}{59263665912}a^{10}+\frac{12556691}{448967166}a^{5}+\frac{5679428}{74827861}$, $\frac{1}{770427656856}a^{11}-\frac{2395045}{224483583}a^{6}+\frac{454646594}{972762193}a$, $\frac{1}{11\!\cdots\!80}a^{12}+\frac{1}{148159164780}a^{10}-\frac{1}{60}a^{9}+\frac{1}{30}a^{8}-\frac{17290025981}{642023047380}a^{7}-\frac{16571493}{748278610}a^{5}+\frac{1}{10}a^{4}+\frac{1}{5}a^{3}+\frac{163651371721}{695524967995}a^{2}+\frac{2}{5}a+\frac{11358856}{374139305}$, $\frac{1}{14\!\cdots\!40}a^{13}+\frac{1}{1926069142140}a^{11}-\frac{1}{148159164780}a^{10}-\frac{1}{60}a^{9}+\frac{303721497709}{8346299615940}a^{8}-\frac{31329407}{1496557220}a^{6}-\frac{29128184}{374139305}a^{5}-\frac{1}{10}a^{4}-\frac{3150322096533}{18083649167870}a^{3}-\frac{1}{5}a^{2}+\frac{1882055381}{4863810965}a+\frac{12693801}{74827861}$, $\frac{1}{20\!\cdots\!20}a^{14}-\frac{1}{1926069142140}a^{11}-\frac{6373318195043}{11\!\cdots\!20}a^{9}+\frac{1}{30}a^{8}+\frac{1}{60}a^{7}-\frac{55667501}{4489671660}a^{6}-\frac{61017999433717}{25\!\cdots\!10}a^{4}-\frac{1}{5}a^{3}-\frac{2}{5}a^{2}+\frac{2008993391}{4863810965}a+\frac{1}{5}$
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
$C_{165}$, which has order $165$ (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: | $7$ | 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: |
$\frac{1419857}{51\!\cdots\!80}a^{14}-\frac{13243}{954816676063536}a^{13}+\frac{779}{24482478873424}a^{12}+\frac{289}{3852138284280}a^{11}-\frac{479295499}{11\!\cdots\!20}a^{9}+\frac{8173787}{4173149807970}a^{8}-\frac{480811}{107003841230}a^{7}-\frac{13721}{1496557220}a^{6}-\frac{1776485237183}{12\!\cdots\!05}a^{4}+\frac{203222155713}{18083649167870}a^{3}-\frac{30205336704}{695524967995}a^{2}-\frac{902792582}{4863810965}a+\frac{28}{5}$, $\frac{66733279}{10\!\cdots\!60}a^{14}+\frac{225131}{477408338031768}a^{13}+\frac{5453}{3060309859178}a^{12}+\frac{6647}{1926069142140}a^{11}-\frac{22526888453}{23\!\cdots\!40}a^{9}-\frac{138954379}{2086574903985}a^{8}-\frac{13462708}{53501920615}a^{7}-\frac{315583}{748278610}a^{6}-\frac{83494806147601}{25\!\cdots\!10}a^{4}-\frac{3454776647121}{9041824583935}a^{3}-\frac{1691498855424}{695524967995}a^{2}-\frac{41528458772}{4863810965}a+\frac{38}{5}$, $\frac{8077}{14815916478}a^{10}-\frac{17544757}{224483583}a^{5}-\frac{26251534559}{74827861}$, $\frac{38\!\cdots\!91}{10\!\cdots\!60}a^{14}+\frac{16\!\cdots\!57}{11\!\cdots\!20}a^{13}-\frac{75\!\cdots\!67}{220342309860816}a^{12}-\frac{21\!\cdots\!28}{53501920615}a^{11}+\frac{20\!\cdots\!61}{148159164780}a^{10}+\frac{27\!\cdots\!57}{397840281693140}a^{9}+\frac{11\!\cdots\!51}{4173149807970}a^{8}-\frac{40\!\cdots\!89}{642023047380}a^{7}-\frac{10\!\cdots\!37}{1496557220}a^{6}+\frac{56\!\cdots\!59}{2244835830}a^{5}+\frac{30\!\cdots\!37}{25\!\cdots\!10}a^{4}+\frac{16\!\cdots\!77}{3616729833574}a^{3}-\frac{75\!\cdots\!22}{695524967995}a^{2}-\frac{60\!\cdots\!94}{4863810965}a+\frac{16\!\cdots\!02}{374139305}$, $\frac{26\!\cdots\!63}{34\!\cdots\!20}a^{14}-\frac{10\!\cdots\!11}{14\!\cdots\!40}a^{13}+\frac{12\!\cdots\!67}{3852138284280}a^{12}+\frac{25\!\cdots\!63}{963034571070}a^{11}-\frac{83\!\cdots\!23}{98772776520}a^{10}+\frac{14\!\cdots\!72}{99460070423285}a^{9}-\frac{27\!\cdots\!51}{2086574903985}a^{8}+\frac{44\!\cdots\!69}{748278610}a^{7}+\frac{22\!\cdots\!83}{4489671660}a^{6}-\frac{58\!\cdots\!74}{374139305}a^{5}+\frac{32\!\cdots\!16}{12\!\cdots\!05}a^{4}-\frac{40\!\cdots\!51}{18083649167870}a^{3}+\frac{49\!\cdots\!79}{4863810965}a^{2}+\frac{41\!\cdots\!89}{4863810965}a-\frac{20\!\cdots\!11}{74827861}$, $\frac{45\!\cdots\!39}{20\!\cdots\!20}a^{14}-\frac{14\!\cdots\!07}{14\!\cdots\!40}a^{13}-\frac{12\!\cdots\!59}{220342309860816}a^{12}+\frac{13\!\cdots\!93}{3852138284280}a^{11}-\frac{62\!\cdots\!81}{296318329560}a^{10}-\frac{38\!\cdots\!63}{11\!\cdots\!20}a^{9}+\frac{13\!\cdots\!99}{8346299615940}a^{8}+\frac{90\!\cdots\!69}{214007682460}a^{7}-\frac{23\!\cdots\!87}{4489671660}a^{6}+\frac{35\!\cdots\!79}{1122417915}a^{5}-\frac{11\!\cdots\!58}{12\!\cdots\!05}a^{4}-\frac{19\!\cdots\!31}{3616729833574}a^{3}+\frac{42\!\cdots\!26}{695524967995}a^{2}-\frac{10\!\cdots\!28}{4863810965}a-\frac{21\!\cdots\!66}{374139305}$, $\frac{65\!\cdots\!43}{10\!\cdots\!60}a^{14}+\frac{62\!\cdots\!67}{367237183101360}a^{13}-\frac{17\!\cdots\!51}{275427887326020}a^{12}-\frac{54\!\cdots\!99}{87548597370}a^{11}+\frac{96\!\cdots\!21}{296318329560}a^{10}+\frac{94\!\cdots\!91}{795680563386280}a^{9}+\frac{20\!\cdots\!21}{642023047380}a^{8}-\frac{75\!\cdots\!63}{642023047380}a^{7}-\frac{51\!\cdots\!29}{4489671660}a^{6}+\frac{13\!\cdots\!79}{2244835830}a^{5}+\frac{52\!\cdots\!41}{25\!\cdots\!10}a^{4}+\frac{75\!\cdots\!47}{1391049935990}a^{3}-\frac{14\!\cdots\!59}{695524967995}a^{2}-\frac{19\!\cdots\!40}{972762193}a+\frac{38\!\cdots\!92}{374139305}$
| 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: | \( 49247044495643600 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 49247044495643600 \cdot 165}{2\cdot\sqrt{5617330134094061678662632074276610010758000000000}}\cr\approx \mathstrut & 1.32543434654911 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 145860*x^10 - 243079056*x^5 - 4696791102432)
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 - 145860*x^10 - 243079056*x^5 - 4696791102432, 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 - 145860*x^10 - 243079056*x^5 - 4696791102432);
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 - 145860*x^10 - 243079056*x^5 - 4696791102432);
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
$S_3\times F_5$ (as 15T11):
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 120 |
| The 15 conjugacy class representatives for $F_5 \times S_3$ |
| Character table for $F_5 \times S_3$ |
Intermediate fields
| 3.1.479116.1, 5.1.148240125.2 |
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 30 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 | R | R | ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{5}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | $15$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.12.8.1 | $x^{12} + 11 x^{9} + 3 x^{8} - 9 x^{6} - 90 x^{5} + 3 x^{4} - 27 x^{3} + 135 x^{2} + 27 x + 55$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ | |
|
\(3\)
| 3.5.4.1 | $x^{5} + 3$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |
| 3.10.8.1 | $x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 598 x^{5} + 750 x^{4} + 640 x^{3} + 280 x^{2} + 40 x + 17$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ | |
|
\(5\)
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 5.12.9.2 | $x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
|
\(11\)
| 11.5.4.5 | $x^{5} + 22$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.10.9.4 | $x^{10} + 22$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
|
\(10889\)
| $\Q_{10889}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |