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Properties

Label 15.1.109...728.1
Degree $15$
Signature $[1, 7]$
Discriminant $-1.096\times 10^{22}$
Root discriminant \(29.47\)
Ramified primes $2,523$
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 - 5*x^13 + 15*x^11 - 2*x^10 + 17*x^9 - 156*x^8 - 156*x^7 + 454*x^6 + 120*x^5 - 1736*x^4 - 668*x^3 - 408*x^2 + 32*x - 8)
 
gp: K = bnfinit(y^15 - 5*y^13 + 15*y^11 - 2*y^10 + 17*y^9 - 156*y^8 - 156*y^7 + 454*y^6 + 120*y^5 - 1736*y^4 - 668*y^3 - 408*y^2 + 32*y - 8, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 5*x^13 + 15*x^11 - 2*x^10 + 17*x^9 - 156*x^8 - 156*x^7 + 454*x^6 + 120*x^5 - 1736*x^4 - 668*x^3 - 408*x^2 + 32*x - 8);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 5*x^13 + 15*x^11 - 2*x^10 + 17*x^9 - 156*x^8 - 156*x^7 + 454*x^6 + 120*x^5 - 1736*x^4 - 668*x^3 - 408*x^2 + 32*x - 8)
 

\( x^{15} - 5 x^{13} + 15 x^{11} - 2 x^{10} + 17 x^{9} - 156 x^{8} - 156 x^{7} + 454 x^{6} + 120 x^{5} + \cdots - 8 \) 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:   \(-10960030903507135433728\) \(\medspace = -\,2^{10}\cdot 523^{7}\) 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:  \(29.47\)
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:  $2^{2/3}523^{1/2}\approx 36.30258142598178$
Ramified primes:   \(2\), \(523\) 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:  \(\Q(\sqrt{-523}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{6}a^{7}+\frac{1}{6}a^{6}+\frac{1}{6}a^{5}+\frac{1}{6}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{12}a^{8}-\frac{1}{12}a^{7}-\frac{1}{12}a^{6}+\frac{5}{12}a^{5}+\frac{1}{6}a^{4}+\frac{1}{6}a^{3}+\frac{1}{6}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{12}a^{9}+\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}a$, $\frac{1}{12}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{108}a^{11}-\frac{1}{108}a^{10}-\frac{1}{27}a^{9}+\frac{1}{36}a^{8}+\frac{2}{27}a^{7}+\frac{13}{54}a^{6}-\frac{19}{108}a^{5}+\frac{1}{54}a^{4}-\frac{2}{27}a^{3}-\frac{1}{2}a^{2}+\frac{2}{9}a+\frac{5}{27}$, $\frac{1}{1080}a^{12}-\frac{1}{540}a^{11}-\frac{13}{360}a^{10}-\frac{1}{54}a^{9}+\frac{1}{216}a^{8}+\frac{1}{20}a^{7}+\frac{11}{120}a^{6}-\frac{19}{45}a^{5}+\frac{229}{540}a^{4}+\frac{13}{54}a^{3}+\frac{7}{18}a^{2}-\frac{127}{270}a-\frac{23}{270}$, $\frac{1}{16200}a^{13}+\frac{1}{8100}a^{12}-\frac{1}{600}a^{11}-\frac{53}{8100}a^{10}+\frac{131}{3240}a^{9}-\frac{17}{2025}a^{8}-\frac{67}{3240}a^{7}+\frac{23}{810}a^{6}-\frac{307}{900}a^{5}+\frac{199}{2025}a^{4}+\frac{119}{810}a^{3}-\frac{1417}{4050}a^{2}+\frac{493}{1350}a-\frac{401}{2025}$, $\frac{1}{108945000}a^{14}-\frac{973}{36315000}a^{13}-\frac{10097}{54472500}a^{12}+\frac{1661}{108945000}a^{11}+\frac{66713}{9078750}a^{10}+\frac{475303}{36315000}a^{9}-\frac{35413}{1008750}a^{8}+\frac{1038559}{21789000}a^{7}-\frac{13068761}{108945000}a^{6}-\frac{24386531}{54472500}a^{5}-\frac{4968317}{18157500}a^{4}+\frac{3731621}{9078750}a^{3}+\frac{1491611}{27236250}a^{2}+\frac{5811007}{13618125}a-\frac{13207483}{27236250}$ 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:  $2$, $3$

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:   $\frac{478409}{108945000}a^{14}+\frac{373879}{108945000}a^{13}-\frac{195383}{9078750}a^{12}-\frac{1871651}{108945000}a^{11}+\frac{3399577}{54472500}a^{10}+\frac{4882531}{108945000}a^{9}+\frac{1073308}{13618125}a^{8}-\frac{13537319}{21789000}a^{7}-\frac{44256583}{36315000}a^{6}+\frac{18824174}{13618125}a^{5}+\frac{108665791}{54472500}a^{4}-\frac{91312579}{13618125}a^{3}-\frac{41753246}{4539375}a^{2}-\frac{66965512}{13618125}a-\frac{5388283}{3026250}$, $\frac{3461}{1345000}a^{14}-\frac{13217}{9078750}a^{13}-\frac{77353}{6052500}a^{12}+\frac{132371}{18157500}a^{11}+\frac{352433}{9078750}a^{10}-\frac{280513}{9078750}a^{9}+\frac{982481}{18157500}a^{8}-\frac{734863}{1815750}a^{7}-\frac{7107917}{36315000}a^{6}+\frac{2627027}{2017500}a^{5}-\frac{4117297}{18157500}a^{4}-\frac{7051769}{1513125}a^{3}+\frac{7410767}{9078750}a^{2}-\frac{7316167}{9078750}a-\frac{81889}{1008750}$, $\frac{100079}{9078750}a^{14}-\frac{228431}{54472500}a^{13}-\frac{3211231}{54472500}a^{12}+\frac{469363}{18157500}a^{11}+\frac{9841619}{54472500}a^{10}-\frac{1446596}{13618125}a^{9}+\frac{4242577}{27236250}a^{8}-\frac{4737646}{2723625}a^{7}-\frac{32012857}{27236250}a^{6}+\frac{28957651}{4539375}a^{5}-\frac{11689237}{13618125}a^{4}-\frac{585291001}{27236250}a^{3}+\frac{23490014}{13618125}a^{2}+\frac{11907362}{4539375}a-\frac{7654042}{13618125}$, $\frac{122801}{18157500}a^{14}-\frac{60863}{36315000}a^{13}-\frac{138157}{4035000}a^{12}+\frac{309047}{36315000}a^{11}+\frac{3734587}{36315000}a^{10}-\frac{1467157}{36315000}a^{9}+\frac{1397339}{12105000}a^{8}-\frac{7721357}{7263000}a^{7}-\frac{28199747}{36315000}a^{6}+\frac{60670463}{18157500}a^{5}-\frac{1999127}{18157500}a^{4}-\frac{6026318}{504375}a^{3}-\frac{3017089}{4539375}a^{2}-\frac{1320137}{1513125}a+\frac{579359}{9078750}$, $\frac{39173}{36315000}a^{14}+\frac{851639}{108945000}a^{13}-\frac{1089611}{108945000}a^{12}-\frac{1470797}{36315000}a^{11}+\frac{4105339}{108945000}a^{10}+\frac{12846521}{108945000}a^{9}-\frac{6809951}{108945000}a^{8}-\frac{453179}{21789000}a^{7}-\frac{19598948}{13618125}a^{6}-\frac{131357}{2017500}a^{5}+\frac{61220039}{13618125}a^{4}-\frac{77830603}{27236250}a^{3}-\frac{204332233}{13618125}a^{2}+\frac{20299597}{9078750}a+\frac{1459499}{13618125}$, $\frac{1211}{7263000}a^{14}-\frac{11507}{21789000}a^{13}-\frac{1453}{5447250}a^{12}+\frac{2041}{7263000}a^{11}+\frac{86089}{10894500}a^{10}-\frac{34253}{21789000}a^{9}-\frac{122501}{10894500}a^{8}-\frac{218503}{4357800}a^{7}+\frac{2340937}{21789000}a^{6}+\frac{64207}{1815750}a^{5}+\frac{4190587}{10894500}a^{4}-\frac{3557473}{2723625}a^{3}-\frac{1751051}{2723625}a^{2}-\frac{275018}{907875}a-\frac{612829}{5447250}$, $\frac{150661}{54472500}a^{14}-\frac{694}{1513125}a^{13}-\frac{145696}{13618125}a^{12}-\frac{2779}{54472500}a^{11}+\frac{204199}{6052500}a^{10}+\frac{211283}{18157500}a^{9}+\frac{731677}{18157500}a^{8}-\frac{2018563}{5447250}a^{7}-\frac{18062771}{54472500}a^{6}+\frac{14295167}{13618125}a^{5}+\frac{3138913}{9078750}a^{4}-\frac{5541298}{1513125}a^{3}-\frac{27423079}{13618125}a^{2}-\frac{10999471}{13618125}a-\frac{597838}{13618125}$ 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:  \( 336028.882665 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 336028.882665 \cdot 2}{2\cdot\sqrt{10960030903507135433728}}\cr\approx \mathstrut & 2.48176006966 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^13 + 15*x^11 - 2*x^10 + 17*x^9 - 156*x^8 - 156*x^7 + 454*x^6 + 120*x^5 - 1736*x^4 - 668*x^3 - 408*x^2 + 32*x - 8)
 
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 - 5*x^13 + 15*x^11 - 2*x^10 + 17*x^9 - 156*x^8 - 156*x^7 + 454*x^6 + 120*x^5 - 1736*x^4 - 668*x^3 - 408*x^2 + 32*x - 8, 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 - 5*x^13 + 15*x^11 - 2*x^10 + 17*x^9 - 156*x^8 - 156*x^7 + 454*x^6 + 120*x^5 - 1736*x^4 - 668*x^3 - 408*x^2 + 32*x - 8);
 
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 - 5*x^13 + 15*x^11 - 2*x^10 + 17*x^9 - 156*x^8 - 156*x^7 + 454*x^6 + 120*x^5 - 1736*x^4 - 668*x^3 - 408*x^2 + 32*x - 8);
 
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

3.1.2092.1, 5.1.273529.1

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 ${\href{/padicField/3.2.0.1}{2} }^{7}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.2.0.1}{2} }^{7}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.5.0.1}{5} }^{3}$ ${\href{/padicField/11.5.0.1}{5} }^{3}$ $15$ $15$ $15$ $15$ $15$ $15$ ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ $15$ $15$ ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ $15$ ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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}$
\(523\) Copy content Toggle raw display $\Q_{523}$$x$$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}$