Properties

Label 15.1.677952124826430464.1
Degree $15$
Signature $[1, 7]$
Discriminant $-6.780\times 10^{17}$
Root discriminant \(15.44\)
Ramified primes $2,131$
Class number $1$
Class group trivial
Galois group $D_{15}$ (as 15T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 + 4*x^13 + 4*x^11 - 2*x^10 - 6*x^9 - 12*x^8 - 25*x^7 + 19*x^6 - 6*x^5 + 28*x^4 + 32*x^3 - 36*x^2 + 4*x + 4)
 
gp: K = bnfinit(y^15 - y^14 + 4*y^13 + 4*y^11 - 2*y^10 - 6*y^9 - 12*y^8 - 25*y^7 + 19*y^6 - 6*y^5 + 28*y^4 + 32*y^3 - 36*y^2 + 4*y + 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - x^14 + 4*x^13 + 4*x^11 - 2*x^10 - 6*x^9 - 12*x^8 - 25*x^7 + 19*x^6 - 6*x^5 + 28*x^4 + 32*x^3 - 36*x^2 + 4*x + 4);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - x^14 + 4*x^13 + 4*x^11 - 2*x^10 - 6*x^9 - 12*x^8 - 25*x^7 + 19*x^6 - 6*x^5 + 28*x^4 + 32*x^3 - 36*x^2 + 4*x + 4)
 

\( x^{15} - x^{14} + 4 x^{13} + 4 x^{11} - 2 x^{10} - 6 x^{9} - 12 x^{8} - 25 x^{7} + 19 x^{6} - 6 x^{5} + 28 x^{4} + 32 x^{3} - 36 x^{2} + 4 x + 4 \) 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:  $[1, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-677952124826430464\) \(\medspace = -\,2^{10}\cdot 131^{7}\) 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:  \(15.44\)
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:  $2^{2/3}131^{1/2}\approx 18.168635476349255$
Ramified primes:   \(2\), \(131\) 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:  \(\Q(\sqrt{-131}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a$, $\frac{1}{232}a^{13}-\frac{9}{232}a^{12}+\frac{19}{232}a^{11}+\frac{13}{232}a^{10}-\frac{23}{232}a^{9}-\frac{37}{232}a^{8}-\frac{23}{232}a^{7}+\frac{19}{232}a^{6}+\frac{45}{116}a^{5}+\frac{7}{116}a^{4}-\frac{7}{58}a^{3}-\frac{3}{29}a^{2}+\frac{10}{29}a-\frac{1}{58}$, $\frac{1}{90712}a^{14}+\frac{35}{22678}a^{13}+\frac{3225}{45356}a^{12}+\frac{4293}{45356}a^{11}+\frac{24}{391}a^{10}-\frac{21}{45356}a^{9}-\frac{8829}{45356}a^{8}+\frac{10215}{45356}a^{7}+\frac{11447}{90712}a^{6}-\frac{10949}{45356}a^{5}+\frac{3092}{11339}a^{4}+\frac{4507}{11339}a^{3}-\frac{7979}{22678}a^{2}-\frac{1048}{11339}a+\frac{5535}{22678}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

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:  $7$
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:   $\frac{16}{493}a^{14}-\frac{33}{1972}a^{13}+\frac{61}{493}a^{12}+\frac{13}{1972}a^{11}+\frac{169}{986}a^{10}-\frac{111}{493}a^{9}-\frac{683}{1972}a^{8}-\frac{2109}{1972}a^{7}-\frac{686}{493}a^{6}-\frac{723}{1972}a^{5}+\frac{387}{1972}a^{4}+\frac{2331}{986}a^{3}+\frac{2105}{986}a^{2}+\frac{1553}{986}a-\frac{108}{493}$, $\frac{111}{45356}a^{14}-\frac{1373}{90712}a^{13}+\frac{449}{90712}a^{12}-\frac{109}{3128}a^{11}-\frac{2201}{90712}a^{10}-\frac{11279}{90712}a^{9}+\frac{2045}{90712}a^{8}-\frac{2095}{90712}a^{7}+\frac{19675}{90712}a^{6}-\frac{893}{22678}a^{5}+\frac{1267}{45356}a^{4}-\frac{16843}{22678}a^{3}-\frac{499}{22678}a^{2}-\frac{1575}{11339}a+\frac{13933}{22678}$, $\frac{1161}{90712}a^{14}-\frac{210}{11339}a^{13}+\frac{2155}{22678}a^{12}-\frac{295}{45356}a^{11}+\frac{2596}{11339}a^{10}+\frac{157}{1564}a^{9}+\frac{21883}{45356}a^{8}+\frac{5277}{45356}a^{7}+\frac{10043}{90712}a^{6}-\frac{8981}{45356}a^{5}-\frac{45583}{45356}a^{4}-\frac{9510}{11339}a^{3}-\frac{23507}{22678}a^{2}+\frac{8339}{22678}a-\frac{8943}{22678}$, $\frac{5303}{45356}a^{14}-\frac{2075}{11339}a^{13}+\frac{6760}{11339}a^{12}-\frac{16423}{45356}a^{11}+\frac{9471}{11339}a^{10}-\frac{8175}{11339}a^{9}+\frac{4517}{45356}a^{8}-\frac{63287}{45356}a^{7}-\frac{72817}{45356}a^{6}+\frac{68987}{22678}a^{5}-\frac{129439}{45356}a^{4}+\frac{104253}{22678}a^{3}-\frac{4091}{11339}a^{2}-\frac{88237}{22678}a+\frac{20458}{11339}$, $\frac{8603}{90712}a^{14}-\frac{65}{1564}a^{13}+\frac{14955}{45356}a^{12}+\frac{5079}{22678}a^{11}+\frac{18815}{45356}a^{10}+\frac{4671}{45356}a^{9}-\frac{27597}{45356}a^{8}-\frac{25195}{22678}a^{7}-\frac{9205}{3128}a^{6}+\frac{17399}{22678}a^{5}-\frac{427}{11339}a^{4}+\frac{27576}{11339}a^{3}+\frac{48600}{11339}a^{2}-\frac{18643}{11339}a-\frac{22617}{22678}$, $\frac{3019}{45356}a^{14}-\frac{687}{11339}a^{13}+\frac{2729}{11339}a^{12}+\frac{2135}{45356}a^{11}+\frac{4261}{22678}a^{10}-\frac{1621}{22678}a^{9}-\frac{25977}{45356}a^{8}-\frac{1029}{1564}a^{7}-\frac{80197}{45356}a^{6}+\frac{29043}{22678}a^{5}-\frac{3407}{45356}a^{4}+\frac{18043}{11339}a^{3}+\frac{30625}{11339}a^{2}-\frac{43161}{22678}a+\frac{2344}{11339}$, $\frac{1346}{11339}a^{14}-\frac{3607}{45356}a^{13}+\frac{19685}{45356}a^{12}+\frac{4239}{22678}a^{11}+\frac{16955}{45356}a^{10}+\frac{3389}{45356}a^{9}-\frac{22971}{22678}a^{8}-\frac{17454}{11339}a^{7}-\frac{179023}{45356}a^{6}+\frac{39581}{22678}a^{5}-\frac{10443}{45356}a^{4}+\frac{40729}{11339}a^{3}+\frac{2408}{391}a^{2}-\frac{81363}{22678}a-\frac{12137}{11339}$ 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:  \( 1343.20606196 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 1343.20606196 \cdot 1}{2\cdot\sqrt{677952124826430464}}\cr\approx \mathstrut & 0.630670051922 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 + 4*x^13 + 4*x^11 - 2*x^10 - 6*x^9 - 12*x^8 - 25*x^7 + 19*x^6 - 6*x^5 + 28*x^4 + 32*x^3 - 36*x^2 + 4*x + 4)
 
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 + 4*x^13 + 4*x^11 - 2*x^10 - 6*x^9 - 12*x^8 - 25*x^7 + 19*x^6 - 6*x^5 + 28*x^4 + 32*x^3 - 36*x^2 + 4*x + 4, 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 + 4*x^13 + 4*x^11 - 2*x^10 - 6*x^9 - 12*x^8 - 25*x^7 + 19*x^6 - 6*x^5 + 28*x^4 + 32*x^3 - 36*x^2 + 4*x + 4);
 
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 + 4*x^13 + 4*x^11 - 2*x^10 - 6*x^9 - 12*x^8 - 25*x^7 + 19*x^6 - 6*x^5 + 28*x^4 + 32*x^3 - 36*x^2 + 4*x + 4);
 
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

$D_{15}$ (as 15T2):

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 9 conjugacy class representatives for $D_{15}$
Character table for $D_{15}$

Intermediate fields

3.1.524.1, 5.1.17161.1

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 R $15$ ${\href{/padicField/5.5.0.1}{5} }^{3}$ $15$ ${\href{/padicField/11.5.0.1}{5} }^{3}$ $15$ ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ $15$ $15$ ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.3.0.1}{3} }^{5}$ $15$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
\(131\) Copy content Toggle raw display $\Q_{131}$$x + 129$$1$$1$$0$Trivial$[\ ]$
131.2.1.2$x^{2} + 131$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 131$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 131$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 131$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 131$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 131$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 131$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.131.2t1.a.a$1$ $ 131 $ \(\Q(\sqrt{-131}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.524.3t2.a.a$2$ $ 2^{2} \cdot 131 $ 3.1.524.1 $S_3$ (as 3T2) $1$ $0$
* 2.131.5t2.a.a$2$ $ 131 $ 5.1.17161.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.131.5t2.a.b$2$ $ 131 $ 5.1.17161.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.524.15t2.a.c$2$ $ 2^{2} \cdot 131 $ 15.1.677952124826430464.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.524.15t2.a.a$2$ $ 2^{2} \cdot 131 $ 15.1.677952124826430464.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.524.15t2.a.b$2$ $ 2^{2} \cdot 131 $ 15.1.677952124826430464.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.524.15t2.a.d$2$ $ 2^{2} \cdot 131 $ 15.1.677952124826430464.1 $D_{15}$ (as 15T2) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.