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Properties

Label 15.1.160...000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-1.608\times 10^{22}$
Root discriminant \(30.23\)
Ramified primes $2,5,7$
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 - 15*x^12 - 20*x^11 - 36*x^10 + 75*x^9 + 200*x^8 + 110*x^7 - 475*x^6 - 480*x^5 + 350*x^4 + 500*x^3 - 1100*x^2 + 500*x - 100)
 
gp: K = bnfinit(y^15 - 15*y^12 - 20*y^11 - 36*y^10 + 75*y^9 + 200*y^8 + 110*y^7 - 475*y^6 - 480*y^5 + 350*y^4 + 500*y^3 - 1100*y^2 + 500*y - 100, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 15*x^12 - 20*x^11 - 36*x^10 + 75*x^9 + 200*x^8 + 110*x^7 - 475*x^6 - 480*x^5 + 350*x^4 + 500*x^3 - 1100*x^2 + 500*x - 100);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 15*x^12 - 20*x^11 - 36*x^10 + 75*x^9 + 200*x^8 + 110*x^7 - 475*x^6 - 480*x^5 + 350*x^4 + 500*x^3 - 1100*x^2 + 500*x - 100)
 

\( x^{15} - 15 x^{12} - 20 x^{11} - 36 x^{10} + 75 x^{9} + 200 x^{8} + 110 x^{7} - 475 x^{6} - 480 x^{5} + \cdots - 100 \) 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:   \(-16084824218750000000000\) \(\medspace = -\,2^{10}\cdot 5^{19}\cdot 7^{7}\) 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:  \(30.23\)
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:  $2^{2/3}5^{13/10}7^{1/2}\approx 34.03272225543917$
Ramified primes:   \(2\), \(5\), \(7\) 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:  \(\Q(\sqrt{-35}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7}a^{9}-\frac{3}{7}a^{8}+\frac{2}{7}a^{7}-\frac{2}{7}a^{6}+\frac{1}{7}a^{5}-\frac{2}{7}a^{4}+\frac{3}{7}a^{3}+\frac{2}{7}a^{2}-\frac{2}{7}$, $\frac{1}{35}a^{10}-\frac{2}{7}a^{7}-\frac{1}{7}a^{6}-\frac{6}{35}a^{5}-\frac{2}{7}a^{4}-\frac{2}{7}a^{3}-\frac{3}{7}a^{2}+\frac{1}{7}a+\frac{3}{7}$, $\frac{1}{35}a^{11}-\frac{2}{7}a^{8}-\frac{1}{7}a^{7}-\frac{6}{35}a^{6}-\frac{2}{7}a^{5}-\frac{2}{7}a^{4}-\frac{3}{7}a^{3}+\frac{1}{7}a^{2}+\frac{3}{7}a$, $\frac{1}{2450}a^{12}+\frac{2}{245}a^{10}+\frac{33}{490}a^{9}+\frac{38}{245}a^{8}+\frac{387}{1225}a^{7}+\frac{25}{98}a^{6}+\frac{78}{245}a^{5}-\frac{32}{245}a^{4}-\frac{47}{98}a^{3}+\frac{101}{245}a^{2}-\frac{5}{49}a-\frac{1}{49}$, $\frac{1}{2450}a^{13}+\frac{2}{245}a^{11}+\frac{1}{98}a^{10}+\frac{3}{245}a^{9}-\frac{313}{1225}a^{8}-\frac{45}{98}a^{7}-\frac{27}{245}a^{6}+\frac{17}{245}a^{5}+\frac{37}{98}a^{4}-\frac{109}{245}a^{3}+\frac{23}{49}a^{2}-\frac{15}{49}a+\frac{3}{7}$, $\frac{1}{57814075900}a^{14}-\frac{266773}{28907037950}a^{13}-\frac{2473783}{14453518975}a^{12}+\frac{46691087}{11562815180}a^{11}+\frac{8900921}{825915370}a^{10}-\frac{595827459}{14453518975}a^{9}-\frac{3016907239}{57814075900}a^{8}-\frac{11753366159}{28907037950}a^{7}+\frac{1927230831}{5781407590}a^{6}-\frac{256681531}{1651830740}a^{5}+\frac{55619131}{1156281518}a^{4}+\frac{98952169}{5781407590}a^{3}+\frac{362742193}{5781407590}a^{2}-\frac{103324541}{578140759}a-\frac{39540189}{578140759}$ 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:  $2$, $7$

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:   $\frac{3649207}{1521423050}a^{14}-\frac{14781}{304284610}a^{13}+\frac{80823}{60856922}a^{12}-\frac{1614103}{43469230}a^{11}-\frac{13916253}{304284610}a^{10}-\frac{162643307}{1521423050}a^{9}+\frac{51246311}{304284610}a^{8}+\frac{26467323}{60856922}a^{7}+\frac{115498489}{304284610}a^{6}-\frac{57479035}{60856922}a^{5}-\frac{339644681}{304284610}a^{4}+\frac{29077891}{60856922}a^{3}+\frac{21865100}{30428461}a^{2}-\frac{60462937}{30428461}a+\frac{6066222}{30428461}$, $\frac{109416107}{57814075900}a^{14}-\frac{217318629}{14453518975}a^{13}-\frac{254419062}{14453518975}a^{12}-\frac{458462919}{11562815180}a^{11}+\frac{541571508}{2890703795}a^{10}+\frac{7333844342}{14453518975}a^{9}+\frac{69748044951}{57814075900}a^{8}+\frac{35138663}{294969775}a^{7}-\frac{22258729469}{5781407590}a^{6}-\frac{83133728079}{11562815180}a^{5}+\frac{4362550649}{2890703795}a^{4}+\frac{88747305049}{5781407590}a^{3}+\frac{67447964707}{5781407590}a^{2}-\frac{4511695495}{578140759}a+\frac{1389543070}{578140759}$, $\frac{19566823}{14453518975}a^{14}+\frac{71210479}{28907037950}a^{13}+\frac{81982309}{28907037950}a^{12}-\frac{56893667}{2890703795}a^{11}-\frac{70737699}{1156281518}a^{10}-\frac{3959453941}{28907037950}a^{9}-\frac{864485767}{14453518975}a^{8}+\frac{8702714691}{28907037950}a^{7}+\frac{594062319}{825915370}a^{6}+\frac{484838321}{2890703795}a^{5}-\frac{7091557259}{5781407590}a^{4}-\frac{7694285967}{5781407590}a^{3}+\frac{1269345674}{2890703795}a^{2}+\frac{11887306}{82591537}a+\frac{28129638}{578140759}$, $\frac{55273637}{57814075900}a^{14}-\frac{16952553}{14453518975}a^{13}+\frac{39123467}{28907037950}a^{12}-\frac{126489337}{11562815180}a^{11}+\frac{3409239}{2890703795}a^{10}-\frac{952125481}{28907037950}a^{9}+\frac{1931698557}{57814075900}a^{8}-\frac{939606396}{14453518975}a^{7}-\frac{125759386}{578140759}a^{6}-\frac{1682724657}{11562815180}a^{5}+\frac{3490752887}{2890703795}a^{4}+\frac{3633916064}{2890703795}a^{3}-\frac{9967962941}{5781407590}a^{2}-\frac{2360871164}{578140759}a+\frac{992232541}{578140759}$, $\frac{15814311}{4129576850}a^{14}+\frac{1456295}{578140759}a^{13}-\frac{57940109}{14453518975}a^{12}-\frac{328954139}{5781407590}a^{11}-\frac{313466173}{2890703795}a^{10}-\frac{2202619836}{14453518975}a^{9}+\frac{1683963327}{5781407590}a^{8}+\frac{15296787334}{14453518975}a^{7}+\frac{1854760226}{2890703795}a^{6}-\frac{12326308077}{5781407590}a^{5}-\frac{8634621652}{2890703795}a^{4}+\frac{497469346}{578140759}a^{3}+\frac{8168060627}{2890703795}a^{2}-\frac{1062440973}{578140759}a+\frac{215071121}{578140759}$, $\frac{181461197}{8259153700}a^{14}+\frac{341194937}{28907037950}a^{13}-\frac{9587166}{2890703795}a^{12}-\frac{3860620079}{11562815180}a^{11}-\frac{3671140371}{5781407590}a^{10}-\frac{2008475003}{2064788425}a^{9}+\frac{77042438551}{57814075900}a^{8}+\frac{4742469531}{825915370}a^{7}+\frac{5744920961}{1156281518}a^{6}-\frac{108164041023}{11562815180}a^{5}-\frac{15098572043}{825915370}a^{4}+\frac{6037808839}{5781407590}a^{3}+\frac{21487697719}{1156281518}a^{2}-\frac{5746034320}{578140759}a+\frac{1144651707}{578140759}$, $\frac{2902467}{578140759}a^{14}-\frac{243877173}{14453518975}a^{13}+\frac{4394199}{578140759}a^{12}-\frac{136177991}{2890703795}a^{11}+\frac{154143251}{2890703795}a^{10}+\frac{529613722}{2890703795}a^{9}+\frac{5172188773}{14453518975}a^{8}-\frac{14596965}{578140759}a^{7}-\frac{7032332694}{2890703795}a^{6}-\frac{1980023733}{2890703795}a^{5}+\frac{3442126938}{578140759}a^{4}-\frac{4243340606}{2890703795}a^{3}-\frac{306405738}{82591537}a^{2}+\frac{335123972}{82591537}a-\frac{386945719}{578140759}$ 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:  \( 913440.648087 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 913440.648087 \cdot 1}{2\cdot\sqrt{16084824218750000000000}}\cr\approx \mathstrut & 2.78439761275 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - 15*x^12 - 20*x^11 - 36*x^10 + 75*x^9 + 200*x^8 + 110*x^7 - 475*x^6 - 480*x^5 + 350*x^4 + 500*x^3 - 1100*x^2 + 500*x - 100)
 
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 - 15*x^12 - 20*x^11 - 36*x^10 + 75*x^9 + 200*x^8 + 110*x^7 - 475*x^6 - 480*x^5 + 350*x^4 + 500*x^3 - 1100*x^2 + 500*x - 100, 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 - 15*x^12 - 20*x^11 - 36*x^10 + 75*x^9 + 200*x^8 + 110*x^7 - 475*x^6 - 480*x^5 + 350*x^4 + 500*x^3 - 1100*x^2 + 500*x - 100);
 
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 - 15*x^12 - 20*x^11 - 36*x^10 + 75*x^9 + 200*x^8 + 110*x^7 - 475*x^6 - 480*x^5 + 350*x^4 + 500*x^3 - 1100*x^2 + 500*x - 100);
 
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

3.1.140.1, 5.1.765625.1

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 $15$ R R $15$ $15$ $15$ ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ $15$ ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.3.0.1}{3} }^{5}$ ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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}$
\(5\) Copy content Toggle raw display 5.5.6.1$x^{5} + 10 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
5.10.13.7$x^{10} + 5 x^{4} + 10$$10$$1$$13$$D_5$$[3/2]_{2}$
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$