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Properties

Label 15.3.334095024862954369.1
Degree $15$
Signature $[3, 6]$
Discriminant $3.341\times 10^{17}$
Root discriminant \(14.73\)
Ramified primes $7,17$
Class number $1$
Class group trivial
Galois group $D_5\times C_3$ (as 15T3)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 3*x^13 - 5*x^12 + x^11 - 19*x^10 + 16*x^9 - 26*x^8 + 55*x^7 - 78*x^6 + 88*x^5 - 54*x^4 + 39*x^3 - 52*x^2 + 21*x - 1)
 
gp: K = bnfinit(y^15 - 2*y^14 + 3*y^13 - 5*y^12 + y^11 - 19*y^10 + 16*y^9 - 26*y^8 + 55*y^7 - 78*y^6 + 88*y^5 - 54*y^4 + 39*y^3 - 52*y^2 + 21*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 2*x^14 + 3*x^13 - 5*x^12 + x^11 - 19*x^10 + 16*x^9 - 26*x^8 + 55*x^7 - 78*x^6 + 88*x^5 - 54*x^4 + 39*x^3 - 52*x^2 + 21*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 2*x^14 + 3*x^13 - 5*x^12 + x^11 - 19*x^10 + 16*x^9 - 26*x^8 + 55*x^7 - 78*x^6 + 88*x^5 - 54*x^4 + 39*x^3 - 52*x^2 + 21*x - 1)
 

\( x^{15} - 2 x^{14} + 3 x^{13} - 5 x^{12} + x^{11} - 19 x^{10} + 16 x^{9} - 26 x^{8} + 55 x^{7} - 78 x^{6} + \cdots - 1 \) Copy content Toggle raw display

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:   \(334095024862954369\) \(\medspace = 7^{12}\cdot 17^{6}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(14.73\)
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:  $7^{5/6}17^{1/2}\approx 20.867615567973587$
Ramified primes:   \(7\), \(17\) Copy content Toggle raw display
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7}a^{9}-\frac{1}{7}a^{7}+\frac{1}{7}a^{6}-\frac{2}{7}a^{5}-\frac{3}{7}a^{4}-\frac{1}{7}a^{3}-\frac{3}{7}a^{2}+\frac{3}{7}a-\frac{1}{7}$, $\frac{1}{7}a^{10}-\frac{1}{7}a^{8}+\frac{1}{7}a^{7}-\frac{2}{7}a^{6}-\frac{3}{7}a^{5}-\frac{1}{7}a^{4}-\frac{3}{7}a^{3}+\frac{3}{7}a^{2}-\frac{1}{7}a$, $\frac{1}{7}a^{11}+\frac{1}{7}a^{8}-\frac{3}{7}a^{7}-\frac{2}{7}a^{6}-\frac{3}{7}a^{5}+\frac{1}{7}a^{4}+\frac{2}{7}a^{3}+\frac{3}{7}a^{2}+\frac{3}{7}a-\frac{1}{7}$, $\frac{1}{7}a^{12}-\frac{3}{7}a^{8}-\frac{1}{7}a^{7}+\frac{3}{7}a^{6}+\frac{3}{7}a^{5}-\frac{2}{7}a^{4}-\frac{3}{7}a^{3}-\frac{1}{7}a^{2}+\frac{3}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{13}-\frac{1}{7}a^{8}-\frac{1}{7}a^{6}-\frac{1}{7}a^{5}+\frac{2}{7}a^{4}+\frac{3}{7}a^{3}+\frac{1}{7}a^{2}+\frac{3}{7}a-\frac{3}{7}$, $\frac{1}{2208935141}a^{14}+\frac{156905185}{2208935141}a^{13}+\frac{140199210}{2208935141}a^{12}+\frac{36539448}{2208935141}a^{11}-\frac{71402612}{2208935141}a^{10}-\frac{109948555}{2208935141}a^{9}+\frac{231484205}{2208935141}a^{8}+\frac{51920276}{315562163}a^{7}-\frac{203771023}{2208935141}a^{6}-\frac{198868886}{2208935141}a^{5}+\frac{1005114245}{2208935141}a^{4}+\frac{374781055}{2208935141}a^{3}+\frac{397619944}{2208935141}a^{2}-\frac{1047472032}{2208935141}a-\frac{538962771}{2208935141}$ Copy content Toggle raw display

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 $2$) Copy content Toggle raw display
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:   $\frac{34796749}{2208935141}a^{14}+\frac{63071795}{2208935141}a^{13}-\frac{86108704}{2208935141}a^{12}+\frac{84797679}{2208935141}a^{11}-\frac{392890051}{2208935141}a^{10}-\frac{905322246}{2208935141}a^{9}-\frac{1884288083}{2208935141}a^{8}-\frac{44267348}{315562163}a^{7}-\frac{625010744}{2208935141}a^{6}+\frac{2256343034}{2208935141}a^{5}-\frac{3841303037}{2208935141}a^{4}+\frac{4312154271}{2208935141}a^{3}+\frac{919713390}{2208935141}a^{2}-\frac{1255045132}{2208935141}a-\frac{3545238008}{2208935141}$, $\frac{48710311}{315562163}a^{14}-\frac{82013641}{315562163}a^{13}+\frac{128597389}{315562163}a^{12}-\frac{216522843}{315562163}a^{11}+\frac{212379}{45080309}a^{10}-\frac{954496707}{315562163}a^{9}+\frac{64843152}{45080309}a^{8}-\frac{1265184821}{315562163}a^{7}+\frac{2328694817}{315562163}a^{6}-\frac{3269922340}{315562163}a^{5}+\frac{3586463521}{315562163}a^{4}-\frac{256273689}{45080309}a^{3}+\frac{1767428062}{315562163}a^{2}-\frac{289166928}{45080309}a+\frac{246247030}{315562163}$, $\frac{5700061}{45080309}a^{14}-\frac{67185092}{315562163}a^{13}+\frac{103262728}{315562163}a^{12}-\frac{171327400}{315562163}a^{11}-\frac{7909361}{315562163}a^{10}-\frac{776415737}{315562163}a^{9}+\frac{384082323}{315562163}a^{8}-\frac{1016475415}{315562163}a^{7}+\frac{1843409354}{315562163}a^{6}-\frac{2589857714}{315562163}a^{5}+\frac{2866941625}{315562163}a^{4}-\frac{1426404346}{315562163}a^{3}+\frac{1385802217}{315562163}a^{2}-\frac{1609571834}{315562163}a+\frac{652818627}{315562163}$, $\frac{356491850}{2208935141}a^{14}-\frac{572280699}{2208935141}a^{13}+\frac{870527907}{2208935141}a^{12}-\frac{1584994221}{2208935141}a^{11}-\frac{47231703}{2208935141}a^{10}-\frac{7141812198}{2208935141}a^{9}+\frac{3244700397}{2208935141}a^{8}-\frac{171234215}{45080309}a^{7}+\frac{18412473428}{2208935141}a^{6}-\frac{21674293955}{2208935141}a^{5}+\frac{26483556556}{2208935141}a^{4}-\frac{14055339958}{2208935141}a^{3}+\frac{15968057022}{2208935141}a^{2}-\frac{18292832899}{2208935141}a+\frac{2176049088}{2208935141}$, $\frac{284244736}{2208935141}a^{14}-\frac{520788040}{2208935141}a^{13}+\frac{712468109}{2208935141}a^{12}-\frac{1263732863}{2208935141}a^{11}+\frac{22603834}{2208935141}a^{10}-\frac{5288860714}{2208935141}a^{9}+\frac{3818393823}{2208935141}a^{8}-\frac{800680957}{315562163}a^{7}+\frac{15009433804}{2208935141}a^{6}-\frac{18818810984}{2208935141}a^{5}+\frac{20926370715}{2208935141}a^{4}-\frac{9979420386}{2208935141}a^{3}+\frac{8807319569}{2208935141}a^{2}-\frac{12688719766}{2208935141}a+\frac{3201360453}{2208935141}$, $\frac{34796749}{2208935141}a^{14}+\frac{63071795}{2208935141}a^{13}-\frac{86108704}{2208935141}a^{12}+\frac{84797679}{2208935141}a^{11}-\frac{392890051}{2208935141}a^{10}-\frac{905322246}{2208935141}a^{9}-\frac{1884288083}{2208935141}a^{8}-\frac{44267348}{315562163}a^{7}-\frac{625010744}{2208935141}a^{6}+\frac{2256343034}{2208935141}a^{5}-\frac{3841303037}{2208935141}a^{4}+\frac{4312154271}{2208935141}a^{3}+\frac{919713390}{2208935141}a^{2}-\frac{1255045132}{2208935141}a-\frac{1336302867}{2208935141}$, $\frac{553916156}{2208935141}a^{14}-\frac{814591636}{2208935141}a^{13}+\frac{1215810328}{2208935141}a^{12}-\frac{2081999847}{2208935141}a^{11}-\frac{617313189}{2208935141}a^{10}-\frac{10705606487}{2208935141}a^{9}+\frac{3124584207}{2208935141}a^{8}-\frac{1788986939}{315562163}a^{7}+\frac{23393183706}{2208935141}a^{6}-\frac{30652065420}{2208935141}a^{5}+\frac{30774562174}{2208935141}a^{4}-\frac{12514592019}{2208935141}a^{3}+\frac{13596901860}{2208935141}a^{2}-\frac{21063011880}{2208935141}a+\frac{407879985}{2208935141}$, $\frac{199374699}{2208935141}a^{14}-\frac{288943861}{2208935141}a^{13}+\frac{453091887}{2208935141}a^{12}-\frac{772294482}{2208935141}a^{11}-\frac{146495005}{2208935141}a^{10}-\frac{3979024613}{2208935141}a^{9}+\frac{1115019548}{2208935141}a^{8}-\frac{717673700}{315562163}a^{7}+\frac{8378973730}{2208935141}a^{6}-\frac{12330600332}{2208935141}a^{5}+\frac{11564692263}{2208935141}a^{4}-\frac{7036534188}{2208935141}a^{3}+\frac{5830661647}{2208935141}a^{2}-\frac{8938603865}{2208935141}a+\frac{1173219093}{2208935141}$ Copy content Toggle raw display
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:  \( 678.185779099 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 678.185779099 \cdot 1}{2\cdot\sqrt{334095024862954369}}\cr\approx \mathstrut & 0.288770534060 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 3*x^13 - 5*x^12 + x^11 - 19*x^10 + 16*x^9 - 26*x^8 + 55*x^7 - 78*x^6 + 88*x^5 - 54*x^4 + 39*x^3 - 52*x^2 + 21*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 - 2*x^14 + 3*x^13 - 5*x^12 + x^11 - 19*x^10 + 16*x^9 - 26*x^8 + 55*x^7 - 78*x^6 + 88*x^5 - 54*x^4 + 39*x^3 - 52*x^2 + 21*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 - 2*x^14 + 3*x^13 - 5*x^12 + x^11 - 19*x^10 + 16*x^9 - 26*x^8 + 55*x^7 - 78*x^6 + 88*x^5 - 54*x^4 + 39*x^3 - 52*x^2 + 21*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 - 2*x^14 + 3*x^13 - 5*x^12 + x^11 - 19*x^10 + 16*x^9 - 26*x^8 + 55*x^7 - 78*x^6 + 88*x^5 - 54*x^4 + 39*x^3 - 52*x^2 + 21*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

$C_3\times D_5$ (as 15T3):

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 30
The 12 conjugacy class representatives for $D_5\times C_3$
Character table for $D_5\times C_3$

Intermediate fields

\(\Q(\zeta_{7})^+\), 5.1.14161.1

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

Galois closure: deg 30
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 $15$ $15$ $15$ R ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }$ ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ R ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ $15$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }$ ${\href{/padicField/41.5.0.1}{5} }^{3}$ ${\href{/padicField/43.5.0.1}{5} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }$ $15$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display 7.3.2.2$x^{3} + 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.2$x^{6} + 42$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} + 42$$6$$1$$5$$C_6$$[\ ]_{6}$
\(17\) Copy content Toggle raw display 17.3.0.1$x^{3} + x + 14$$1$$3$$0$$C_3$$[\ ]^{3}$
17.6.3.2$x^{6} + 289 x^{2} - 68782$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} + 289 x^{2} - 68782$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$