Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 6*x^13 + 29*x^12 - 66*x^11 - 32*x^10 + 34*x^9 + 100*x^8 + 122*x^7 + 56*x^6 + 12*x^5 + 11*x^4 - 9*x^3 + 7*x^2 - 6*x + 1)
gp: K = bnfinit(y^15 - y^14 - 6*y^13 + 29*y^12 - 66*y^11 - 32*y^10 + 34*y^9 + 100*y^8 + 122*y^7 + 56*y^6 + 12*y^5 + 11*y^4 - 9*y^3 + 7*y^2 - 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - x^14 - 6*x^13 + 29*x^12 - 66*x^11 - 32*x^10 + 34*x^9 + 100*x^8 + 122*x^7 + 56*x^6 + 12*x^5 + 11*x^4 - 9*x^3 + 7*x^2 - 6*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - x^14 - 6*x^13 + 29*x^12 - 66*x^11 - 32*x^10 + 34*x^9 + 100*x^8 + 122*x^7 + 56*x^6 + 12*x^5 + 11*x^4 - 9*x^3 + 7*x^2 - 6*x + 1)
\( x^{15} - x^{14} - 6 x^{13} + 29 x^{12} - 66 x^{11} - 32 x^{10} + 34 x^{9} + 100 x^{8} + 122 x^{7} + \cdots + 1 \)
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: | $[5, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-695671349660522996651\)
\(\medspace = -\,11^{13}\cdot 67^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(24.52\) | 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: | $11^{9/10}67^{2/3}\approx 142.76988787487005$ | ||
| Ramified primes: |
\(11\), \(67\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{22839775770703}a^{14}-\frac{6903584901936}{22839775770703}a^{13}+\frac{9631903651088}{22839775770703}a^{12}+\frac{247050083791}{993033729161}a^{11}-\frac{5220525109199}{22839775770703}a^{10}-\frac{163225509968}{22839775770703}a^{9}-\frac{10632550631013}{22839775770703}a^{8}+\frac{349164712002}{22839775770703}a^{7}-\frac{4487861335841}{22839775770703}a^{6}-\frac{447315490048}{993033729161}a^{5}-\frac{5098693346993}{22839775770703}a^{4}-\frac{1922047733774}{22839775770703}a^{3}-\frac{364824634455}{993033729161}a^{2}-\frac{8058628160423}{22839775770703}a-\frac{6683229636798}{22839775770703}$
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: | $9$ | 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: |
$\frac{6351689519994}{22839775770703}a^{14}-\frac{6209379615875}{22839775770703}a^{13}-\frac{35727781392263}{22839775770703}a^{12}+\frac{7860385976066}{993033729161}a^{11}-\frac{431619045224481}{22839775770703}a^{10}-\frac{136587174847217}{22839775770703}a^{9}+\frac{51197148590444}{22839775770703}a^{8}+\frac{511786645210981}{22839775770703}a^{7}+\frac{10\!\cdots\!05}{22839775770703}a^{6}+\frac{25861528112965}{993033729161}a^{5}+\frac{266477323939550}{22839775770703}a^{4}+\frac{185029972342046}{22839775770703}a^{3}-\frac{3601011516035}{993033729161}a^{2}+\frac{81295760012165}{22839775770703}a-\frac{29540336465516}{22839775770703}$, $\frac{2993219547924}{22839775770703}a^{14}-\frac{3383869978704}{22839775770703}a^{13}-\frac{18062132655528}{22839775770703}a^{12}+\frac{3983358975662}{993033729161}a^{11}-\frac{206515038915245}{22839775770703}a^{10}-\frac{98532673309578}{22839775770703}a^{9}+\frac{196696711330556}{22839775770703}a^{8}+\frac{218253645806816}{22839775770703}a^{7}+\frac{126577261693965}{22839775770703}a^{6}+\frac{6509635947663}{993033729161}a^{5}+\frac{262439361931046}{22839775770703}a^{4}+\frac{300746504300285}{22839775770703}a^{3}+\frac{5117274059900}{993033729161}a^{2}+\frac{14750801530831}{22839775770703}a-\frac{23530739596972}{22839775770703}$, $\frac{2568785348885}{22839775770703}a^{14}-\frac{4036750161952}{22839775770703}a^{13}-\frac{11731863981068}{22839775770703}a^{12}+\frac{3540590552889}{993033729161}a^{11}-\frac{226231433310827}{22839775770703}a^{10}+\frac{77896089698415}{22839775770703}a^{9}+\frac{639229512000}{22839775770703}a^{8}+\frac{94976135136499}{22839775770703}a^{7}+\frac{282032132151573}{22839775770703}a^{6}+\frac{7673977307639}{993033729161}a^{5}+\frac{202258109210977}{22839775770703}a^{4}+\frac{183663488734262}{22839775770703}a^{3}+\frac{141300385234}{993033729161}a^{2}+\frac{45404488498942}{22839775770703}a-\frac{53664161867748}{22839775770703}$, $\frac{2568785348885}{22839775770703}a^{14}-\frac{4036750161952}{22839775770703}a^{13}-\frac{11731863981068}{22839775770703}a^{12}+\frac{3540590552889}{993033729161}a^{11}-\frac{226231433310827}{22839775770703}a^{10}+\frac{77896089698415}{22839775770703}a^{9}+\frac{639229512000}{22839775770703}a^{8}+\frac{94976135136499}{22839775770703}a^{7}+\frac{282032132151573}{22839775770703}a^{6}+\frac{7673977307639}{993033729161}a^{5}+\frac{202258109210977}{22839775770703}a^{4}+\frac{183663488734262}{22839775770703}a^{3}+\frac{141300385234}{993033729161}a^{2}+\frac{45404488498942}{22839775770703}a-\frac{30824386097045}{22839775770703}$, $a$, $\frac{11206975181181}{22839775770703}a^{14}-\frac{5786308125006}{22839775770703}a^{13}-\frac{73155991392699}{22839775770703}a^{12}+\frac{12729562989359}{993033729161}a^{11}-\frac{579942517246614}{22839775770703}a^{10}-\frac{729659892098005}{22839775770703}a^{9}+\frac{239965555659315}{22839775770703}a^{8}+\frac{13\!\cdots\!89}{22839775770703}a^{7}+\frac{19\!\cdots\!20}{22839775770703}a^{6}+\frac{57617863272244}{993033729161}a^{5}+\frac{321165612688275}{22839775770703}a^{4}-\frac{19622883087041}{22839775770703}a^{3}-\frac{6757880952103}{993033729161}a^{2}+\frac{8774015893008}{22839775770703}a+\frac{11224285172613}{22839775770703}$, $\frac{781149740705}{22839775770703}a^{14}+\frac{10060706246000}{22839775770703}a^{13}-\frac{11056479290958}{22839775770703}a^{12}-\frac{2279534020888}{993033729161}a^{11}+\frac{241983546178969}{22839775770703}a^{10}-\frac{574620659395967}{22839775770703}a^{9}-\frac{783758410476420}{22839775770703}a^{8}+\frac{661705239259953}{22839775770703}a^{7}+\frac{15\!\cdots\!74}{22839775770703}a^{6}+\frac{59411765486074}{993033729161}a^{5}+\frac{528970242273809}{22839775770703}a^{4}+\frac{104965491631530}{22839775770703}a^{3}+\frac{4191414041293}{993033729161}a^{2}+\frac{55156252866122}{22839775770703}a-\frac{72714188609437}{22839775770703}$, $\frac{2899341071993}{22839775770703}a^{14}-\frac{6531220561566}{22839775770703}a^{13}-\frac{13476887243843}{22839775770703}a^{12}+\frac{4613927037126}{993033729161}a^{11}-\frac{298879794664084}{22839775770703}a^{10}+\frac{153173173650001}{22839775770703}a^{9}+\frac{208266721315769}{22839775770703}a^{8}+\frac{119656375750665}{22839775770703}a^{7}+\frac{19808988410459}{22839775770703}a^{6}-\frac{15892079438160}{993033729161}a^{5}+\frac{38729897630110}{22839775770703}a^{4}+\frac{9749715653887}{22839775770703}a^{3}+\frac{6121332292157}{993033729161}a^{2}+\frac{25358733422465}{22839775770703}a-\frac{199627993956}{22839775770703}$, $\frac{3419981067944}{22839775770703}a^{14}+\frac{8361977347027}{22839775770703}a^{13}-\frac{45024934287006}{22839775770703}a^{12}+\frac{1798681645301}{993033729161}a^{11}+\frac{200637842045621}{22839775770703}a^{10}-\frac{12\!\cdots\!90}{22839775770703}a^{9}+\frac{536676101433546}{22839775770703}a^{8}+\frac{14\!\cdots\!78}{22839775770703}a^{7}+\frac{426466839582153}{22839775770703}a^{6}+\frac{17408352966585}{993033729161}a^{5}-\frac{60013796711151}{22839775770703}a^{4}-\frac{216637649097048}{22839775770703}a^{3}+\frac{11741873306502}{993033729161}a^{2}-\frac{181119407498358}{22839775770703}a+\frac{21982052666971}{22839775770703}$
| 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: | \( 47044.8303705 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 47044.8303705 \cdot 1}{2\cdot\sqrt{695671349660522996651}}\cr\approx \mathstrut & 0.279466124543 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 6*x^13 + 29*x^12 - 66*x^11 - 32*x^10 + 34*x^9 + 100*x^8 + 122*x^7 + 56*x^6 + 12*x^5 + 11*x^4 - 9*x^3 + 7*x^2 - 6*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 - x^14 - 6*x^13 + 29*x^12 - 66*x^11 - 32*x^10 + 34*x^9 + 100*x^8 + 122*x^7 + 56*x^6 + 12*x^5 + 11*x^4 - 9*x^3 + 7*x^2 - 6*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 - x^14 - 6*x^13 + 29*x^12 - 66*x^11 - 32*x^10 + 34*x^9 + 100*x^8 + 122*x^7 + 56*x^6 + 12*x^5 + 11*x^4 - 9*x^3 + 7*x^2 - 6*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 - x^14 - 6*x^13 + 29*x^12 - 66*x^11 - 32*x^10 + 34*x^9 + 100*x^8 + 122*x^7 + 56*x^6 + 12*x^5 + 11*x^4 - 9*x^3 + 7*x^2 - 6*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
$C_7^3:C_6$ (as 15T44):
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 2430 |
| The 39 conjugacy class representatives for $C_7^3:C_6$ |
| Character table for $C_7^3:C_6$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\) |
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 15 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 45 siblings: | data not computed |
| Minimal sibling: | 15.5.154972454814106259.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.5.0.1}{5} }$ | ${\href{/padicField/3.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | R | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{9}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | $15$ | $15$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ | $15$ | $15$ | $15$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.10.9.7 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
|
\(67\)
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 67.3.0.1 | $x^{3} + 6 x + 65$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 67.3.0.1 | $x^{3} + 6 x + 65$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 67.3.2.2 | $x^{3} + 268$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 67.3.2.3 | $x^{3} + 134$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |