LMFDB - Number field 15.1.7245968805874891233103.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 16*x^13 - 24*x^12 + 38*x^11 + x^10 + 24*x^9 + 3*x^8 + 52*x^7 - 20*x^6 + 13*x^5 + 100*x^4 - 18*x^3 - 21*x^2 + 83*x + 1)

gp: K = bnfinit(y^15 - 6*y^14 + 16*y^13 - 24*y^12 + 38*y^11 + y^10 + 24*y^9 + 3*y^8 + 52*y^7 - 20*y^6 + 13*y^5 + 100*y^4 - 18*y^3 - 21*y^2 + 83*y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 6*x^14 + 16*x^13 - 24*x^12 + 38*x^11 + x^10 + 24*x^9 + 3*x^8 + 52*x^7 - 20*x^6 + 13*x^5 + 100*x^4 - 18*x^3 - 21*x^2 + 83*x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 6*x^14 + 16*x^13 - 24*x^12 + 38*x^11 + x^10 + 24*x^9 + 3*x^8 + 52*x^7 - 20*x^6 + 13*x^5 + 100*x^4 - 18*x^3 - 21*x^2 + 83*x + 1)

$ x^{15} - 6 x^{14} + 16 x^{13} - 24 x^{12} + 38 x^{11} + x^{10} + 24 x^{9} + 3 x^{8} + 52 x^{7} + \cdots + 1 $ x^15 - 6*x^14 + 16*x^13 - 24*x^12 + 38*x^11 + x^10 + 24*x^9 + 3*x^8 + 52*x^7 - 20*x^6 + 13*x^5 + 100*x^4 - 18*x^3 - 21*x^2 + 83*x + 1

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:	$-7245968805874891233103$ -7245968805874891233103 $\medspace = -\,1327^{7}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$28.66$	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:	$1327^{1/2}\approx 36.42801120017397$
Ramified primes:	$1327$ 1327	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-1327}) $
$\card{ \Aut(K/\Q) }$:	$1$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{15}a^{8}-\frac{2}{15}a^{7}-\frac{7}{15}a^{6}+\frac{4}{15}a^{5}+\frac{2}{15}a^{4}+\frac{7}{15}a^{3}-\frac{7}{15}a^{2}+\frac{4}{15}a+\frac{4}{15}$, $\frac{1}{15}a^{9}-\frac{1}{15}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{15}a^{4}+\frac{2}{15}a^{3}-\frac{1}{3}a^{2}+\frac{7}{15}a-\frac{2}{15}$, $\frac{1}{45}a^{10}+\frac{1}{45}a^{9}+\frac{2}{45}a^{7}-\frac{2}{45}a^{6}-\frac{2}{9}a^{5}+\frac{2}{9}a^{4}-\frac{1}{45}a^{3}+\frac{19}{45}a+\frac{7}{45}$, $\frac{1}{45}a^{11}-\frac{1}{45}a^{9}-\frac{1}{45}a^{8}+\frac{2}{45}a^{7}+\frac{13}{45}a^{6}+\frac{8}{45}a^{5}-\frac{17}{45}a^{4}-\frac{4}{9}a^{3}-\frac{1}{9}a^{2}+\frac{7}{15}a-\frac{19}{45}$, $\frac{1}{45}a^{12}-\frac{1}{45}a^{8}+\frac{2}{15}a^{7}-\frac{1}{15}a^{6}-\frac{1}{5}a^{5}-\frac{1}{45}a^{4}+\frac{1}{15}a^{3}+\frac{4}{15}a^{2}+\frac{2}{5}a+\frac{2}{9}$, $\frac{1}{2025}a^{13}-\frac{19}{2025}a^{12}-\frac{4}{405}a^{11}-\frac{4}{675}a^{10}+\frac{49}{2025}a^{9}+\frac{1}{135}a^{8}-\frac{178}{2025}a^{7}+\frac{187}{2025}a^{6}+\frac{37}{81}a^{5}+\frac{464}{2025}a^{4}-\frac{719}{2025}a^{3}-\frac{37}{405}a^{2}+\frac{979}{2025}a+\frac{187}{2025}$, $\frac{1}{115425}a^{14}+\frac{1}{6075}a^{13}+\frac{833}{115425}a^{12}-\frac{1087}{115425}a^{11}-\frac{632}{115425}a^{10}+\frac{3182}{115425}a^{9}+\frac{1427}{115425}a^{8}+\frac{7463}{115425}a^{7}-\frac{15368}{38475}a^{6}-\frac{47276}{115425}a^{5}+\frac{16823}{115425}a^{4}-\frac{3529}{38475}a^{3}+\frac{5078}{38475}a^{2}-\frac{1022}{38475}a+\frac{42791}{115425}$ 1, a, a^2, a^3, a^4, a^5, a^6, 1/3*a^7 - 1/3*a^6 + 1/3*a^5 - 1/3*a^4 + 1/3*a^3 - 1/3*a^2 + 1/3*a - 1/3, 1/15*a^8 - 2/15*a^7 - 7/15*a^6 + 4/15*a^5 + 2/15*a^4 + 7/15*a^3 - 7/15*a^2 + 4/15*a + 4/15, 1/15*a^9 - 1/15*a^7 - 1/3*a^6 + 1/3*a^5 + 1/15*a^4 + 2/15*a^3 - 1/3*a^2 + 7/15*a - 2/15, 1/45*a^10 + 1/45*a^9 + 2/45*a^7 - 2/45*a^6 - 2/9*a^5 + 2/9*a^4 - 1/45*a^3 + 19/45*a + 7/45, 1/45*a^11 - 1/45*a^9 - 1/45*a^8 + 2/45*a^7 + 13/45*a^6 + 8/45*a^5 - 17/45*a^4 - 4/9*a^3 - 1/9*a^2 + 7/15*a - 19/45, 1/45*a^12 - 1/45*a^8 + 2/15*a^7 - 1/15*a^6 - 1/5*a^5 - 1/45*a^4 + 1/15*a^3 + 4/15*a^2 + 2/5*a + 2/9, 1/2025*a^13 - 19/2025*a^12 - 4/405*a^11 - 4/675*a^10 + 49/2025*a^9 + 1/135*a^8 - 178/2025*a^7 + 187/2025*a^6 + 37/81*a^5 + 464/2025*a^4 - 719/2025*a^3 - 37/405*a^2 + 979/2025*a + 187/2025, 1/115425*a^14 + 1/6075*a^13 + 833/115425*a^12 - 1087/115425*a^11 - 632/115425*a^10 + 3182/115425*a^9 + 1427/115425*a^8 + 7463/115425*a^7 - 15368/38475*a^6 - 47276/115425*a^5 + 16823/115425*a^4 - 3529/38475*a^3 + 5078/38475*a^2 - 1022/38475*a + 42791/115425

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$3$

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 $ -1 (order $2$)	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:	$\frac{433}{23085}a^{14}-\frac{682}{6075}a^{13}+\frac{35362}{115425}a^{12}-\frac{2330}{4617}a^{11}+\frac{104201}{115425}a^{10}-\frac{37382}{115425}a^{9}+\frac{19562}{23085}a^{8}+\frac{25654}{115425}a^{7}+\frac{6913}{38475}a^{6}+\frac{3617}{4617}a^{5}+\frac{10993}{115425}a^{4}+\frac{3506}{4275}a^{3}+\frac{385}{513}a^{2}-\frac{4184}{38475}a+\frac{105979}{115425}$, $\frac{1597}{115425}a^{14}-\frac{97}{1215}a^{13}+\frac{22208}{115425}a^{12}-\frac{26509}{115425}a^{11}+\frac{42517}{115425}a^{10}+\frac{8882}{115425}a^{9}+\frac{56774}{115425}a^{8}-\frac{8357}{23085}a^{7}+\frac{2768}{4275}a^{6}+\frac{53443}{115425}a^{5}-\frac{22336}{115425}a^{4}-\frac{29939}{38475}a^{3}+\frac{35896}{38475}a^{2}-\frac{199}{1425}a+\frac{49346}{115425}$, $\frac{2039}{115425}a^{14}-\frac{664}{6075}a^{13}+\frac{6464}{23085}a^{12}-\frac{40133}{115425}a^{11}+\frac{53531}{115425}a^{10}+\frac{655}{4617}a^{9}+\frac{7783}{115425}a^{8}-\frac{47372}{115425}a^{7}+\frac{5183}{7695}a^{6}+\frac{51941}{115425}a^{5}-\frac{15836}{115425}a^{4}+\frac{2111}{2565}a^{3}-\frac{2821}{12825}a^{2}-\frac{14}{38475}a+\frac{29636}{23085}$, $\frac{3127}{115425}a^{14}-\frac{112}{1215}a^{13}+\frac{8783}{115425}a^{12}+\frac{12971}{115425}a^{11}+\frac{23752}{115425}a^{10}+\frac{187097}{115425}a^{9}+\frac{287264}{115425}a^{8}+\frac{64918}{23085}a^{7}+\frac{113957}{38475}a^{6}+\frac{343438}{115425}a^{5}+\frac{73409}{115425}a^{4}+\frac{5989}{4275}a^{3}+\frac{46277}{12825}a^{2}+\frac{44162}{38475}a-\frac{103954}{115425}$, $\frac{4654}{115425}a^{14}-\frac{1391}{6075}a^{13}+\frac{67187}{115425}a^{12}-\frac{99328}{115425}a^{11}+\frac{173437}{115425}a^{10}+\frac{13823}{115425}a^{9}+\frac{175748}{115425}a^{8}+\frac{44882}{115425}a^{7}+\frac{65243}{38475}a^{6}-\frac{85154}{115425}a^{5}-\frac{89308}{115425}a^{4}+\frac{3512}{1425}a^{3}-\frac{22276}{12825}a^{2}-\frac{44008}{38475}a+\frac{109304}{115425}$, $\frac{356}{115425}a^{14}-\frac{79}{6075}a^{13}+\frac{2143}{115425}a^{12}-\frac{1082}{115425}a^{11}+\frac{10133}{115425}a^{10}-\frac{3218}{115425}a^{9}+\frac{45457}{115425}a^{8}-\frac{6782}{115425}a^{7}+\frac{6502}{38475}a^{6}-\frac{7846}{115425}a^{5}+\frac{45598}{115425}a^{4}-\frac{13283}{12825}a^{3}+\frac{3551}{12825}a^{2}+\frac{23863}{38475}a-\frac{14189}{115425}$, $a$ 433/23085a^14 - 682/6075a^13 + 35362/115425a^12 - 2330/4617a^11 + 104201/115425a^10 - 37382/115425a^9 + 19562/23085a^8 + 25654/115425a^7 + 6913/38475a^6 + 3617/4617a^5 + 10993/115425a^4 + 3506/4275a^3 + 385/513a^2 - 4184/38475a + 105979/115425, 1597/115425a^14 - 97/1215a^13 + 22208/115425a^12 - 26509/115425a^11 + 42517/115425a^10 + 8882/115425a^9 + 56774/115425a^8 - 8357/23085a^7 + 2768/4275a^6 + 53443/115425a^5 - 22336/115425a^4 - 29939/38475a^3 + 35896/38475a^2 - 199/1425a + 49346/115425, 2039/115425a^14 - 664/6075a^13 + 6464/23085a^12 - 40133/115425a^11 + 53531/115425a^10 + 655/4617a^9 + 7783/115425a^8 - 47372/115425a^7 + 5183/7695a^6 + 51941/115425a^5 - 15836/115425a^4 + 2111/2565a^3 - 2821/12825a^2 - 14/38475a + 29636/23085, 3127/115425a^14 - 112/1215a^13 + 8783/115425a^12 + 12971/115425a^11 + 23752/115425a^10 + 187097/115425a^9 + 287264/115425a^8 + 64918/23085a^7 + 113957/38475a^6 + 343438/115425a^5 + 73409/115425a^4 + 5989/4275a^3 + 46277/12825a^2 + 44162/38475a - 103954/115425, 4654/115425a^14 - 1391/6075a^13 + 67187/115425a^12 - 99328/115425a^11 + 173437/115425a^10 + 13823/115425a^9 + 175748/115425a^8 + 44882/115425a^7 + 65243/38475a^6 - 85154/115425a^5 - 89308/115425a^4 + 3512/1425a^3 - 22276/12825a^2 - 44008/38475a + 109304/115425, 356/115425a^14 - 79/6075a^13 + 2143/115425a^12 - 1082/115425a^11 + 10133/115425a^10 - 3218/115425a^9 + 45457/115425a^8 - 6782/115425a^7 + 6502/38475a^6 - 7846/115425a^5 + 45598/115425a^4 - 13283/12825a^3 + 3551/12825a^2 + 23863/38475a - 14189/115425, a	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:	$ 225706.053096 $	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 225706.053096 \cdot 1}{2\cdot\sqrt{7245968805874891233103}}\cr\approx \mathstrut & 1.02507125409 \end{aligned}\]

# 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 - 24*x^12 + 38*x^11 + x^10 + 24*x^9 + 3*x^8 + 52*x^7 - 20*x^6 + 13*x^5 + 100*x^4 - 18*x^3 - 21*x^2 + 83*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 - 6*x^14 + 16*x^13 - 24*x^12 + 38*x^11 + x^10 + 24*x^9 + 3*x^8 + 52*x^7 - 20*x^6 + 13*x^5 + 100*x^4 - 18*x^3 - 21*x^2 + 83*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 - 6*x^14 + 16*x^13 - 24*x^12 + 38*x^11 + x^10 + 24*x^9 + 3*x^8 + 52*x^7 - 20*x^6 + 13*x^5 + 100*x^4 - 18*x^3 - 21*x^2 + 83*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 - 6*x^14 + 16*x^13 - 24*x^12 + 38*x^11 + x^10 + 24*x^9 + 3*x^8 + 52*x^7 - 20*x^6 + 13*x^5 + 100*x^4 - 18*x^3 - 21*x^2 + 83*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

$D_{15}$ (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

3.1.1327.1, 5.1.1760929.1

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	$15$	${\href{/padicField/3.2.0.1}{2} }^{7}{,}\,{\href{/padicField/3.1.0.1}{1} }$	${\href{/padicField/5.2.0.1}{2} }^{7}{,}\,{\href{/padicField/5.1.0.1}{1} }$	${\href{/padicField/7.2.0.1}{2} }^{7}{,}\,{\href{/padicField/7.1.0.1}{1} }$	${\href{/padicField/11.3.0.1}{3} }^{5}$	$15$	${\href{/padicField/17.5.0.1}{5} }^{3}$	${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$	${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$	$15$	${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$	${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$	${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$	${\href{/padicField/43.5.0.1}{5} }^{3}$	$15$	${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$	${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$1327$ 1327	$\Q_{1327}$	$x$	$1$	$1$	$0$	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 $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

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