Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 13*x^12 - 18*x^11 + 38*x^9 - 58*x^8 + 38*x^7 - 12*x^6 - 58*x^5 + 110*x^4 - 77*x^3 + 22*x^2 - 1)
gp: K = bnfinit(y^15 - 2*y^14 + 13*y^12 - 18*y^11 + 38*y^9 - 58*y^8 + 38*y^7 - 12*y^6 - 58*y^5 + 110*y^4 - 77*y^3 + 22*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 2*x^14 + 13*x^12 - 18*x^11 + 38*x^9 - 58*x^8 + 38*x^7 - 12*x^6 - 58*x^5 + 110*x^4 - 77*x^3 + 22*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 2*x^14 + 13*x^12 - 18*x^11 + 38*x^9 - 58*x^8 + 38*x^7 - 12*x^6 - 58*x^5 + 110*x^4 - 77*x^3 + 22*x^2 - 1)
\( x^{15} - 2 x^{14} + 13 x^{12} - 18 x^{11} + 38 x^{9} - 58 x^{8} + 38 x^{7} - 12 x^{6} - 58 x^{5} + \cdots - 1 \)
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: | $[3, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1850284719998930944\)
\(\medspace = 2^{12}\cdot 277^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.51\) | 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: | $2^{4/5}277^{1/2}\approx 28.97769793904893$ | ||
| Ramified primes: |
\(2\), \(277\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{1556527306}a^{14}+\frac{99626645}{778263653}a^{13}-\frac{283355867}{1556527306}a^{12}+\frac{349066895}{1556527306}a^{11}+\frac{277800451}{1556527306}a^{10}-\frac{764406965}{1556527306}a^{9}-\frac{4811803}{1556527306}a^{8}-\frac{467901591}{1556527306}a^{7}+\frac{24713537}{1556527306}a^{6}-\frac{307221051}{1556527306}a^{5}+\frac{475411645}{1556527306}a^{4}-\frac{367866759}{1556527306}a^{3}+\frac{216306077}{778263653}a^{2}-\frac{684773075}{1556527306}a-\frac{58242151}{778263653}$
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: | $8$ | 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: |
$\frac{829425391}{1556527306}a^{14}-\frac{494120054}{778263653}a^{13}-\frac{845462091}{1556527306}a^{12}+\frac{10176571595}{1556527306}a^{11}-\frac{6675917235}{1556527306}a^{10}-\frac{6020577975}{1556527306}a^{9}+\frac{27393551201}{1556527306}a^{8}-\frac{25892779527}{1556527306}a^{7}+\frac{8700428711}{1556527306}a^{6}+\frac{21526391}{1556527306}a^{5}-\frac{50170903409}{1556527306}a^{4}+\frac{51068950123}{1556527306}a^{3}-\frac{9227026198}{778263653}a^{2}-\frac{3358350237}{1556527306}a+\frac{1065296088}{778263653}$, $\frac{610910645}{778263653}a^{14}-\frac{1616828317}{1556527306}a^{13}-\frac{677355752}{778263653}a^{12}+\frac{15343395921}{1556527306}a^{11}-\frac{11351990153}{1556527306}a^{10}-\frac{11026375761}{1556527306}a^{9}+\frac{41856362257}{1556527306}a^{8}-\frac{40413899255}{1556527306}a^{7}+\frac{9616690173}{1556527306}a^{6}+\frac{2121822173}{1556527306}a^{5}-\frac{72615982651}{1556527306}a^{4}+\frac{85228734051}{1556527306}a^{3}-\frac{19477963819}{1556527306}a^{2}-\frac{4147334581}{778263653}a+\frac{3405316207}{1556527306}$, $\frac{335305337}{778263653}a^{14}-\frac{845462091}{1556527306}a^{13}-\frac{302979244}{778263653}a^{12}+\frac{8253739803}{1556527306}a^{11}-\frac{6020577975}{1556527306}a^{10}-\frac{4124613657}{1556527306}a^{9}+\frac{22213893151}{1556527306}a^{8}-\frac{22817736147}{1556527306}a^{7}+\frac{9974631083}{1556527306}a^{6}-\frac{2064230731}{1556527306}a^{5}-\frac{40167842887}{1556527306}a^{4}+\frac{45411702711}{1556527306}a^{3}-\frac{21605708839}{1556527306}a^{2}+\frac{1065296088}{778263653}a+\frac{829425391}{1556527306}$, $\frac{71219581}{778263653}a^{14}-\frac{534683019}{1556527306}a^{13}+\frac{135833346}{778263653}a^{12}+\frac{2202670265}{1556527306}a^{11}-\frac{5610762069}{1556527306}a^{10}+\frac{1589560787}{1556527306}a^{9}+\frac{8155576689}{1556527306}a^{8}-\frac{16153550975}{1556527306}a^{7}+\frac{11949715347}{1556527306}a^{6}-\frac{1532831331}{1556527306}a^{5}-\frac{9456654285}{1556527306}a^{4}+\frac{30735553145}{1556527306}a^{3}-\frac{22676768167}{1556527306}a^{2}+\frac{1379204312}{778263653}a+\frac{1755720209}{1556527306}$, $\frac{648030691}{1556527306}a^{14}-\frac{794472385}{1556527306}a^{13}-\frac{811608079}{1556527306}a^{12}+\frac{3996286073}{778263653}a^{11}-\frac{2575840498}{778263653}a^{10}-\frac{3229496318}{778263653}a^{9}+\frac{10405882238}{778263653}a^{8}-\frac{9499309331}{778263653}a^{7}+\frac{1553833024}{778263653}a^{6}+\frac{156369869}{778263653}a^{5}-\frac{18614310377}{778263653}a^{4}+\frac{20408168262}{778263653}a^{3}-\frac{4882719661}{1556527306}a^{2}-\frac{3724827965}{1556527306}a+\frac{772782681}{1556527306}$, $\frac{1514041457}{1556527306}a^{14}-\frac{2937051925}{1556527306}a^{13}-\frac{320568985}{1556527306}a^{12}+\frac{9948243222}{778263653}a^{11}-\frac{12963961651}{778263653}a^{10}-\frac{1698166396}{778263653}a^{9}+\frac{29582803494}{778263653}a^{8}-\frac{41601670321}{778263653}a^{7}+\frac{23800981099}{778263653}a^{6}-\frac{4530253747}{778263653}a^{5}-\frac{45586006921}{778263653}a^{4}+\frac{80763156803}{778263653}a^{3}-\frac{97241744751}{1556527306}a^{2}+\frac{13969522091}{1556527306}a+\frac{3370298231}{1556527306}$, $\frac{148888205}{778263653}a^{14}+\frac{431109343}{1556527306}a^{13}-\frac{724247100}{778263653}a^{12}+\frac{2756224891}{1556527306}a^{11}+\frac{7615038093}{1556527306}a^{10}-\frac{10432718197}{1556527306}a^{9}+\frac{1888507051}{1556527306}a^{8}+\frac{17921278027}{1556527306}a^{7}-\frac{24363003085}{1556527306}a^{6}+\frac{5591279367}{1556527306}a^{5}-\frac{16704710491}{1556527306}a^{4}-\frac{27371184219}{1556527306}a^{3}+\frac{53592317227}{1556527306}a^{2}-\frac{5009908417}{778263653}a-\frac{4064407685}{1556527306}$, $\frac{396960276}{778263653}a^{14}-\frac{725478884}{778263653}a^{13}-\frac{68406976}{778263653}a^{12}+\frac{5049447346}{778263653}a^{11}-\frac{6300116447}{778263653}a^{10}-\frac{340500843}{778263653}a^{9}+\frac{14180156985}{778263653}a^{8}-\frac{20799699932}{778263653}a^{7}+\frac{13734485539}{778263653}a^{6}-\frac{5266256534}{778263653}a^{5}-\frac{22297458289}{778263653}a^{4}+\frac{39852014225}{778263653}a^{3}-\frac{27611783596}{778263653}a^{2}+\frac{9464108570}{778263653}a-\frac{1510056955}{778263653}$
| 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: | \( 2752.96294871 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 2752.96294871 \cdot 1}{2\cdot\sqrt{1850284719998930944}}\cr\approx \mathstrut & 0.498104295182 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 13*x^12 - 18*x^11 + 38*x^9 - 58*x^8 + 38*x^7 - 12*x^6 - 58*x^5 + 110*x^4 - 77*x^3 + 22*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 - 2*x^14 + 13*x^12 - 18*x^11 + 38*x^9 - 58*x^8 + 38*x^7 - 12*x^6 - 58*x^5 + 110*x^4 - 77*x^3 + 22*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 - 2*x^14 + 13*x^12 - 18*x^11 + 38*x^9 - 58*x^8 + 38*x^7 - 12*x^6 - 58*x^5 + 110*x^4 - 77*x^3 + 22*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 - 2*x^14 + 13*x^12 - 18*x^11 + 38*x^9 - 58*x^8 + 38*x^7 - 12*x^6 - 58*x^5 + 110*x^4 - 77*x^3 + 22*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
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 120 |
| The 7 conjugacy class representatives for $S_5$ |
| Character table for $S_5$ |
Intermediate fields
| 5.1.4432.1 |
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: | 5.1.4432.1 |
| Degree 6 sibling: | 6.2.340062928.2 |
| Degree 10 siblings: | data not computed |
| Degree 12 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 40 sibling: | data not computed |
| Minimal sibling: | 5.1.4432.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{3}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.5.0.1}{5} }^{3}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.5.0.1}{5} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }^{5}$ | ${\href{/padicField/23.3.0.1}{3} }^{5}$ | ${\href{/padicField/29.5.0.1}{5} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }^{3}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.5.0.1}{5} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.5.4.1 | $x^{5} + 2$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |
| 2.10.8.1 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 55 x^{5} + 55 x^{4} + 10 x^{3} - 25 x^{2} - 5 x + 7$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ | |
|
\(277\)
| $\Q_{277}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |