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Properties

Label 15.5.154972454814106259.1
Degree $15$
Signature $[5, 5]$
Discriminant $-1.550\times 10^{17}$
Root discriminant \(14.00\)
Ramified primes $11,67$
Class number $1$
Class group trivial
Galois group $C_7^3:C_6$ (as 15T44)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 2*x^13 + x^12 + x^11 - x^10 + 2*x^9 - 15*x^8 + 8*x^7 + 28*x^6 - 34*x^5 - 7*x^4 + 27*x^3 - 7*x^2 - 3*x + 1)
 
gp: K = bnfinit(y^15 - y^14 - 2*y^13 + y^12 + y^11 - y^10 + 2*y^9 - 15*y^8 + 8*y^7 + 28*y^6 - 34*y^5 - 7*y^4 + 27*y^3 - 7*y^2 - 3*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - x^14 - 2*x^13 + x^12 + x^11 - x^10 + 2*x^9 - 15*x^8 + 8*x^7 + 28*x^6 - 34*x^5 - 7*x^4 + 27*x^3 - 7*x^2 - 3*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - x^14 - 2*x^13 + x^12 + x^11 - x^10 + 2*x^9 - 15*x^8 + 8*x^7 + 28*x^6 - 34*x^5 - 7*x^4 + 27*x^3 - 7*x^2 - 3*x + 1)
 

\( x^{15} - x^{14} - 2 x^{13} + x^{12} + x^{11} - x^{10} + 2 x^{9} - 15 x^{8} + 8 x^{7} + 28 x^{6} + \cdots + 1 \) 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:  $[5, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-154972454814106259\) \(\medspace = -\,11^{13}\cdot 67^{2}\) 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:  \(14.00\)
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:   \(11\), \(67\) 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:  \(\Q(\sqrt{-11}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1375967}a^{14}+\frac{175735}{1375967}a^{13}-\frac{613357}{1375967}a^{12}+\frac{221128}{1375967}a^{11}+\frac{90195}{1375967}a^{10}-\frac{631321}{1375967}a^{9}-\frac{232077}{1375967}a^{8}-\frac{621807}{1375967}a^{7}-\frac{79672}{1375967}a^{6}+\frac{601628}{1375967}a^{5}-\frac{230139}{1375967}a^{4}+\frac{90720}{1375967}a^{3}-\frac{559682}{1375967}a^{2}+\frac{597135}{1375967}a-\frac{6898}{1375967}$ 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:  $9$
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:   $\frac{9871118}{1375967}a^{14}-\frac{5442576}{1375967}a^{13}-\frac{22352934}{1375967}a^{12}+\frac{195017}{1375967}a^{11}+\frac{10168561}{1375967}a^{10}-\frac{5609891}{1375967}a^{9}+\frac{16901521}{1375967}a^{8}-\frac{140513656}{1375967}a^{7}+\frac{15248762}{1375967}a^{6}+\frac{286186857}{1375967}a^{5}-\frac{211351518}{1375967}a^{4}-\frac{166885954}{1375967}a^{3}+\frac{198831449}{1375967}a^{2}+\frac{18484429}{1375967}a-\frac{24636408}{1375967}$, $\frac{2954835}{1375967}a^{14}-\frac{1377570}{1375967}a^{13}-\frac{6917210}{1375967}a^{12}-\frac{439608}{1375967}a^{11}+\frac{3046529}{1375967}a^{10}-\frac{1390323}{1375967}a^{9}+\frac{5191165}{1375967}a^{8}-\frac{41374953}{1375967}a^{7}+\frac{707811}{1375967}a^{6}+\frac{87320377}{1375967}a^{5}-\frac{57031774}{1375967}a^{4}-\frac{55546486}{1375967}a^{3}+\frac{57685609}{1375967}a^{2}+\frac{8146219}{1375967}a-\frac{7132494}{1375967}$, $\frac{1450324}{1375967}a^{14}-\frac{431204}{1375967}a^{13}-\frac{3712168}{1375967}a^{12}-\frac{390954}{1375967}a^{11}+\frac{1542424}{1375967}a^{10}-\frac{645425}{1375967}a^{9}+\frac{2204592}{1375967}a^{8}-\frac{19723503}{1375967}a^{7}-\frac{3384903}{1375967}a^{6}+\frac{46596970}{1375967}a^{5}-\frac{25687417}{1375967}a^{4}-\frac{33722369}{1375967}a^{3}+\frac{30118715}{1375967}a^{2}+\frac{5591940}{1375967}a-\frac{3806796}{1375967}$, $a$, $\frac{5494597}{1375967}a^{14}-\frac{2846191}{1375967}a^{13}-\frac{12822567}{1375967}a^{12}+\frac{113142}{1375967}a^{11}+\frac{5893959}{1375967}a^{10}-\frac{3138561}{1375967}a^{9}+\frac{9205248}{1375967}a^{8}-\frac{77583185}{1375967}a^{7}+\frac{5248701}{1375967}a^{6}+\frac{164217103}{1375967}a^{5}-\frac{117463343}{1375967}a^{4}-\frac{100794874}{1375967}a^{3}+\frac{115621493}{1375967}a^{2}+\frac{10484392}{1375967}a-\frac{15854728}{1375967}$, $\frac{13996050}{1375967}a^{14}-\frac{8460232}{1375967}a^{13}-\frac{31332111}{1375967}a^{12}+\frac{1378211}{1375967}a^{11}+\frac{14821072}{1375967}a^{10}-\frac{8036830}{1375967}a^{9}+\frac{24834469}{1375967}a^{8}-\frac{200082968}{1375967}a^{7}+\frac{33016177}{1375967}a^{6}+\frac{404291653}{1375967}a^{5}-\frac{312370017}{1375967}a^{4}-\frac{224334716}{1375967}a^{3}+\frac{287797026}{1375967}a^{2}+\frac{20588044}{1375967}a-\frac{35803487}{1375967}$, $\frac{5483797}{1375967}a^{14}-\frac{3325698}{1375967}a^{13}-\frac{12472105}{1375967}a^{12}+\frac{609454}{1375967}a^{11}+\frac{5972595}{1375967}a^{10}-\frac{2788246}{1375967}a^{9}+\frac{10000941}{1375967}a^{8}-\frac{78162512}{1375967}a^{7}+\frac{12606761}{1375967}a^{6}+\frac{161198943}{1375967}a^{5}-\frac{122462413}{1375967}a^{4}-\frac{88498667}{1375967}a^{3}+\frac{114188095}{1375967}a^{2}+\frac{7831787}{1375967}a-\frac{12906612}{1375967}$, $\frac{10978394}{1375967}a^{14}-\frac{6171889}{1375967}a^{13}-\frac{25294672}{1375967}a^{12}+\frac{722596}{1375967}a^{11}+\frac{11866554}{1375967}a^{10}-\frac{5926807}{1375967}a^{9}+\frac{19206189}{1375967}a^{8}-\frac{155745697}{1375967}a^{7}+\frac{17855462}{1375967}a^{6}+\frac{325416046}{1375967}a^{5}-\frac{239925756}{1375967}a^{4}-\frac{189293541}{1375967}a^{3}+\frac{229809588}{1375967}a^{2}+\frac{18316179}{1375967}a-\frac{28761340}{1375967}$, $\frac{17550747}{1375967}a^{14}-\frac{10049003}{1375967}a^{13}-\frac{39734123}{1375967}a^{12}+\frac{1252205}{1375967}a^{11}+\frac{18022284}{1375967}a^{10}-\frac{10259379}{1375967}a^{9}+\frac{30689452}{1375967}a^{8}-\frac{249594030}{1375967}a^{7}+\frac{32980535}{1375967}a^{6}+\frac{511898243}{1375967}a^{5}-\frac{386785103}{1375967}a^{4}-\frac{288151214}{1375967}a^{3}+\frac{363226421}{1375967}a^{2}+\frac{23216490}{1375967}a-\frac{43251288}{1375967}$ 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:  \( 492.79424719 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 492.79424719 \cdot 1}{2\cdot\sqrt{154972454814106259}}\cr\approx \mathstrut & 0.19613617318 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 2*x^13 + x^12 + x^11 - x^10 + 2*x^9 - 15*x^8 + 8*x^7 + 28*x^6 - 34*x^5 - 7*x^4 + 27*x^3 - 7*x^2 - 3*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 - 2*x^13 + x^12 + x^11 - x^10 + 2*x^9 - 15*x^8 + 8*x^7 + 28*x^6 - 34*x^5 - 7*x^4 + 27*x^3 - 7*x^2 - 3*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 - 2*x^13 + x^12 + x^11 - x^10 + 2*x^9 - 15*x^8 + 8*x^7 + 28*x^6 - 34*x^5 - 7*x^4 + 27*x^3 - 7*x^2 - 3*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 - 2*x^13 + x^12 + x^11 - x^10 + 2*x^9 - 15*x^8 + 8*x^7 + 28*x^6 - 34*x^5 - 7*x^4 + 27*x^3 - 7*x^2 - 3*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: 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 ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.5.0.1}{5} }$ ${\href{/padicField/3.5.0.1}{5} }^{3}$ $15$ ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ R ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ $15$ $15$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ $15$ $15$ $15$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(11\) Copy content Toggle raw display 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}$
\(67\) Copy content Toggle raw display 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}$
67.3.0.1$x^{3} + 6 x + 65$$1$$3$$0$$C_3$$[\ ]^{3}$
67.3.2.1$x^{3} + 67$$3$$1$$2$$C_3$$[\ ]_{3}$
67.3.0.1$x^{3} + 6 x + 65$$1$$3$$0$$C_3$$[\ ]^{3}$