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Properties

Label 15.1.669...847.1
Degree $15$
Signature $[1, 7]$
Discriminant $-6.691\times 10^{22}$
Root discriminant \(33.24\)
Ramified prime $1823$
Class number $2$
Class group [2]
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 - 6*x^14 + 16*x^13 - 41*x^12 + 67*x^11 - 53*x^10 + 156*x^9 - 529*x^8 + 588*x^7 + 168*x^6 - 271*x^5 - 1281*x^4 + 1423*x^3 - 254*x^2 + 152*x - 200)
 
gp: K = bnfinit(y^15 - 6*y^14 + 16*y^13 - 41*y^12 + 67*y^11 - 53*y^10 + 156*y^9 - 529*y^8 + 588*y^7 + 168*y^6 - 271*y^5 - 1281*y^4 + 1423*y^3 - 254*y^2 + 152*y - 200, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 6*x^14 + 16*x^13 - 41*x^12 + 67*x^11 - 53*x^10 + 156*x^9 - 529*x^8 + 588*x^7 + 168*x^6 - 271*x^5 - 1281*x^4 + 1423*x^3 - 254*x^2 + 152*x - 200);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 6*x^14 + 16*x^13 - 41*x^12 + 67*x^11 - 53*x^10 + 156*x^9 - 529*x^8 + 588*x^7 + 168*x^6 - 271*x^5 - 1281*x^4 + 1423*x^3 - 254*x^2 + 152*x - 200)
 

\( x^{15} - 6 x^{14} + 16 x^{13} - 41 x^{12} + 67 x^{11} - 53 x^{10} + 156 x^{9} - 529 x^{8} + 588 x^{7} + \cdots - 200 \) 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:   \(-66912411284534354746847\) \(\medspace = -\,1823^{7}\) 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:  \(33.24\)
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:  $1823^{1/2}\approx 42.69660408041839$
Ramified primes:   \(1823\) 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:  \(\Q(\sqrt{-1823}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{244}a^{13}+\frac{15}{244}a^{12}+\frac{53}{244}a^{11}-\frac{12}{61}a^{10}-\frac{59}{244}a^{9}-\frac{13}{122}a^{8}-\frac{23}{61}a^{7}-\frac{113}{244}a^{6}-\frac{121}{244}a^{5}+\frac{5}{244}a^{4}-\frac{39}{122}a^{3}+\frac{83}{244}a^{2}+\frac{21}{61}a-\frac{23}{61}$, $\frac{1}{25\!\cdots\!04}a^{14}+\frac{44\!\cdots\!23}{25\!\cdots\!04}a^{13}-\frac{27\!\cdots\!21}{25\!\cdots\!04}a^{12}-\frac{37\!\cdots\!11}{12\!\cdots\!52}a^{11}-\frac{17\!\cdots\!87}{25\!\cdots\!04}a^{10}+\frac{59\!\cdots\!92}{16\!\cdots\!69}a^{9}-\frac{60\!\cdots\!41}{64\!\cdots\!76}a^{8}+\frac{25\!\cdots\!43}{25\!\cdots\!04}a^{7}+\frac{41\!\cdots\!19}{25\!\cdots\!04}a^{6}-\frac{99\!\cdots\!41}{25\!\cdots\!04}a^{5}+\frac{454175958362043}{17\!\cdots\!98}a^{4}+\frac{34\!\cdots\!43}{25\!\cdots\!04}a^{3}+\frac{35\!\cdots\!15}{12\!\cdots\!52}a^{2}+\frac{17\!\cdots\!77}{16\!\cdots\!69}a-\frac{15\!\cdots\!89}{32\!\cdots\!38}$ 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

$C_{2}$, which has order $2$

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:   $\frac{17\!\cdots\!75}{25\!\cdots\!04}a^{14}+\frac{21\!\cdots\!41}{25\!\cdots\!04}a^{13}-\frac{45\!\cdots\!19}{25\!\cdots\!04}a^{12}+\frac{62\!\cdots\!29}{12\!\cdots\!52}a^{11}-\frac{39\!\cdots\!89}{25\!\cdots\!04}a^{10}+\frac{17\!\cdots\!29}{64\!\cdots\!76}a^{9}-\frac{10\!\cdots\!01}{64\!\cdots\!76}a^{8}+\frac{10\!\cdots\!81}{25\!\cdots\!04}a^{7}-\frac{53\!\cdots\!51}{25\!\cdots\!04}a^{6}+\frac{73\!\cdots\!77}{25\!\cdots\!04}a^{5}+\frac{15\!\cdots\!83}{35\!\cdots\!96}a^{4}-\frac{49\!\cdots\!99}{25\!\cdots\!04}a^{3}-\frac{56\!\cdots\!53}{12\!\cdots\!52}a^{2}+\frac{10\!\cdots\!45}{16\!\cdots\!69}a-\frac{57\!\cdots\!23}{32\!\cdots\!38}$, $\frac{59\!\cdots\!21}{25\!\cdots\!04}a^{14}-\frac{34\!\cdots\!05}{25\!\cdots\!04}a^{13}+\frac{85\!\cdots\!95}{25\!\cdots\!04}a^{12}-\frac{10\!\cdots\!41}{12\!\cdots\!52}a^{11}+\frac{31\!\cdots\!77}{25\!\cdots\!04}a^{10}-\frac{76\!\cdots\!93}{16\!\cdots\!69}a^{9}+\frac{16\!\cdots\!87}{64\!\cdots\!76}a^{8}-\frac{26\!\cdots\!77}{25\!\cdots\!04}a^{7}+\frac{20\!\cdots\!43}{25\!\cdots\!04}a^{6}+\frac{28\!\cdots\!95}{25\!\cdots\!04}a^{5}-\frac{43\!\cdots\!33}{35\!\cdots\!96}a^{4}-\frac{54\!\cdots\!01}{25\!\cdots\!04}a^{3}+\frac{14\!\cdots\!39}{12\!\cdots\!52}a^{2}+\frac{27\!\cdots\!59}{32\!\cdots\!38}a+\frac{24\!\cdots\!79}{32\!\cdots\!38}$, $\frac{83\!\cdots\!73}{25\!\cdots\!04}a^{14}-\frac{44\!\cdots\!33}{25\!\cdots\!04}a^{13}+\frac{10\!\cdots\!15}{25\!\cdots\!04}a^{12}-\frac{14\!\cdots\!87}{12\!\cdots\!52}a^{11}+\frac{38\!\cdots\!97}{25\!\cdots\!04}a^{10}-\frac{49\!\cdots\!83}{64\!\cdots\!76}a^{9}+\frac{28\!\cdots\!39}{64\!\cdots\!76}a^{8}-\frac{33\!\cdots\!21}{25\!\cdots\!04}a^{7}+\frac{22\!\cdots\!55}{25\!\cdots\!04}a^{6}+\frac{28\!\cdots\!75}{25\!\cdots\!04}a^{5}-\frac{436424929906868}{890839600674449}a^{4}-\frac{71\!\cdots\!45}{25\!\cdots\!04}a^{3}+\frac{13\!\cdots\!77}{12\!\cdots\!52}a^{2}-\frac{64\!\cdots\!82}{16\!\cdots\!69}a+\frac{27\!\cdots\!57}{32\!\cdots\!38}$, $\frac{10\!\cdots\!75}{25\!\cdots\!04}a^{14}-\frac{60\!\cdots\!83}{25\!\cdots\!04}a^{13}+\frac{15\!\cdots\!45}{25\!\cdots\!04}a^{12}-\frac{20\!\cdots\!45}{12\!\cdots\!52}a^{11}+\frac{62\!\cdots\!79}{25\!\cdots\!04}a^{10}-\frac{10\!\cdots\!41}{64\!\cdots\!76}a^{9}+\frac{37\!\cdots\!57}{64\!\cdots\!76}a^{8}-\frac{52\!\cdots\!11}{25\!\cdots\!04}a^{7}+\frac{54\!\cdots\!53}{25\!\cdots\!04}a^{6}+\frac{26\!\cdots\!65}{25\!\cdots\!04}a^{5}-\frac{11\!\cdots\!50}{890839600674449}a^{4}-\frac{14\!\cdots\!87}{25\!\cdots\!04}a^{3}+\frac{77\!\cdots\!63}{12\!\cdots\!52}a^{2}-\frac{13\!\cdots\!43}{16\!\cdots\!69}a-\frac{12\!\cdots\!23}{32\!\cdots\!38}$, $\frac{87\!\cdots\!09}{25\!\cdots\!04}a^{14}-\frac{41\!\cdots\!25}{25\!\cdots\!04}a^{13}+\frac{86\!\cdots\!47}{25\!\cdots\!04}a^{12}-\frac{12\!\cdots\!21}{12\!\cdots\!52}a^{11}+\frac{25\!\cdots\!05}{25\!\cdots\!04}a^{10}-\frac{89\!\cdots\!09}{64\!\cdots\!76}a^{9}+\frac{28\!\cdots\!59}{64\!\cdots\!76}a^{8}-\frac{28\!\cdots\!25}{25\!\cdots\!04}a^{7}+\frac{89\!\cdots\!43}{25\!\cdots\!04}a^{6}+\frac{38\!\cdots\!03}{25\!\cdots\!04}a^{5}+\frac{29571809911665}{35\!\cdots\!96}a^{4}-\frac{91\!\cdots\!45}{25\!\cdots\!04}a^{3}-\frac{58\!\cdots\!59}{12\!\cdots\!52}a^{2}+\frac{12\!\cdots\!13}{32\!\cdots\!38}a+\frac{13\!\cdots\!37}{32\!\cdots\!38}$, $\frac{27638236621839}{16\!\cdots\!69}a^{14}+\frac{630276046967917}{64\!\cdots\!76}a^{13}-\frac{11\!\cdots\!63}{16\!\cdots\!69}a^{12}+\frac{57\!\cdots\!13}{64\!\cdots\!76}a^{11}-\frac{14\!\cdots\!65}{64\!\cdots\!76}a^{10}-\frac{41\!\cdots\!41}{32\!\cdots\!38}a^{9}+\frac{35\!\cdots\!00}{16\!\cdots\!69}a^{8}-\frac{31\!\cdots\!17}{32\!\cdots\!38}a^{7}-\frac{25\!\cdots\!63}{64\!\cdots\!76}a^{6}+\frac{109350775923948}{16\!\cdots\!69}a^{5}+\frac{22\!\cdots\!97}{35\!\cdots\!96}a^{4}-\frac{12\!\cdots\!31}{64\!\cdots\!76}a^{3}-\frac{82\!\cdots\!61}{32\!\cdots\!38}a^{2}-\frac{53\!\cdots\!79}{32\!\cdots\!38}a+\frac{703955263691538}{16\!\cdots\!69}$, $\frac{12\!\cdots\!83}{16\!\cdots\!69}a^{14}-\frac{30\!\cdots\!05}{64\!\cdots\!76}a^{13}+\frac{83\!\cdots\!53}{64\!\cdots\!76}a^{12}-\frac{21\!\cdots\!65}{64\!\cdots\!76}a^{11}+\frac{89\!\cdots\!90}{16\!\cdots\!69}a^{10}-\frac{31\!\cdots\!29}{64\!\cdots\!76}a^{9}+\frac{20\!\cdots\!26}{16\!\cdots\!69}a^{8}-\frac{68\!\cdots\!82}{16\!\cdots\!69}a^{7}+\frac{32\!\cdots\!41}{64\!\cdots\!76}a^{6}+\frac{51\!\cdots\!33}{64\!\cdots\!76}a^{5}-\frac{73\!\cdots\!45}{35\!\cdots\!96}a^{4}-\frac{29\!\cdots\!99}{32\!\cdots\!38}a^{3}+\frac{72\!\cdots\!25}{64\!\cdots\!76}a^{2}-\frac{12\!\cdots\!37}{32\!\cdots\!38}a+\frac{67\!\cdots\!84}{16\!\cdots\!69}$ 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:  \( 164195.256154 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 164195.256154 \cdot 2}{2\cdot\sqrt{66912411284534354746847}}\cr\approx \mathstrut & 0.490790716665 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 16*x^13 - 41*x^12 + 67*x^11 - 53*x^10 + 156*x^9 - 529*x^8 + 588*x^7 + 168*x^6 - 271*x^5 - 1281*x^4 + 1423*x^3 - 254*x^2 + 152*x - 200)
 
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 - 6*x^14 + 16*x^13 - 41*x^12 + 67*x^11 - 53*x^10 + 156*x^9 - 529*x^8 + 588*x^7 + 168*x^6 - 271*x^5 - 1281*x^4 + 1423*x^3 - 254*x^2 + 152*x - 200, 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 - 6*x^14 + 16*x^13 - 41*x^12 + 67*x^11 - 53*x^10 + 156*x^9 - 529*x^8 + 588*x^7 + 168*x^6 - 271*x^5 - 1281*x^4 + 1423*x^3 - 254*x^2 + 152*x - 200);
 
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 - 6*x^14 + 16*x^13 - 41*x^12 + 67*x^11 - 53*x^10 + 156*x^9 - 529*x^8 + 588*x^7 + 168*x^6 - 271*x^5 - 1281*x^4 + 1423*x^3 - 254*x^2 + 152*x - 200);
 
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

3.1.1823.1, 5.1.3323329.1

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 ${\href{/padicField/2.3.0.1}{3} }^{5}$ $15$ ${\href{/padicField/5.2.0.1}{2} }^{7}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.5.0.1}{5} }^{3}$ ${\href{/padicField/11.5.0.1}{5} }^{3}$ ${\href{/padicField/13.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ $15$ ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ $15$ ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(1823\) Copy content Toggle raw display $\Q_{1823}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$