Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 30*x^13 + 360*x^11 - 12*x^10 - 2200*x^9 + 240*x^8 + 7200*x^7 - 1680*x^6 - 11980*x^5 + 4800*x^4 + 7800*x^3 - 4800*x^2 + 400*x + 104)
gp: K = bnfinit(y^15 - 30*y^13 + 360*y^11 - 12*y^10 - 2200*y^9 + 240*y^8 + 7200*y^7 - 1680*y^6 - 11980*y^5 + 4800*y^4 + 7800*y^3 - 4800*y^2 + 400*y + 104, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 30*x^13 + 360*x^11 - 12*x^10 - 2200*x^9 + 240*x^8 + 7200*x^7 - 1680*x^6 - 11980*x^5 + 4800*x^4 + 7800*x^3 - 4800*x^2 + 400*x + 104);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 30*x^13 + 360*x^11 - 12*x^10 - 2200*x^9 + 240*x^8 + 7200*x^7 - 1680*x^6 - 11980*x^5 + 4800*x^4 + 7800*x^3 - 4800*x^2 + 400*x + 104)
\( x^{15} - 30 x^{13} + 360 x^{11} - 12 x^{10} - 2200 x^{9} + 240 x^{8} + 7200 x^{7} - 1680 x^{6} + \cdots + 104 \)
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: | $[15, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1730160900125000000000000\)
\(\medspace = 2^{12}\cdot 5^{15}\cdot 7^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(41.29\) | 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: | $2^{4/5}5^{23/20}7^{5/6}\approx 56.09031360603209$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{4}a^{12}$, $\frac{1}{164}a^{13}-\frac{17}{164}a^{12}+\frac{15}{164}a^{11}-\frac{1}{82}a^{10}+\frac{7}{82}a^{9}-\frac{2}{41}a^{8}+\frac{7}{82}a^{7}-\frac{7}{82}a^{6}+\frac{1}{41}a^{5}+\frac{7}{41}a^{4}+\frac{4}{41}a^{3}-\frac{2}{41}a^{2}-\frac{17}{41}a-\frac{13}{41}$, $\frac{1}{164}a^{14}+\frac{13}{164}a^{12}+\frac{7}{164}a^{11}-\frac{5}{41}a^{10}-\frac{4}{41}a^{9}-\frac{10}{41}a^{8}-\frac{11}{82}a^{7}+\frac{3}{41}a^{6}+\frac{7}{82}a^{5}-\frac{16}{41}a^{3}-\frac{10}{41}a^{2}-\frac{15}{41}a-\frac{16}{41}$
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 $1$ (assuming GRH)
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: | $14$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(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: |
$\frac{1}{4}a^{10}-5a^{8}+35a^{6}-a^{5}-100a^{4}+10a^{3}+100a^{2}-20a-4$, $\frac{1}{4}a^{10}-5a^{8}+35a^{6}-\frac{3}{2}a^{5}-100a^{4}+15a^{3}+100a^{2}-30a-2$, $\frac{3}{41}a^{14}+\frac{1}{82}a^{13}-\frac{329}{164}a^{12}-\frac{91}{164}a^{11}+\frac{1723}{82}a^{10}+\frac{15}{2}a^{9}-\frac{8489}{82}a^{8}-\frac{3357}{82}a^{7}+\frac{19287}{82}a^{6}+\frac{3652}{41}a^{5}-\frac{7899}{41}a^{4}-\frac{2685}{41}a^{3}-\frac{452}{41}a^{2}+\frac{811}{41}a+\frac{69}{41}$, $\frac{3}{41}a^{14}+\frac{1}{82}a^{13}-\frac{329}{164}a^{12}-\frac{91}{164}a^{11}+\frac{3487}{164}a^{10}+\frac{15}{2}a^{9}-\frac{8899}{82}a^{8}-\frac{3357}{82}a^{7}+\frac{22157}{82}a^{6}+\frac{7181}{82}a^{5}-\frac{11999}{41}a^{4}-\frac{2070}{41}a^{3}+\frac{3648}{41}a^{2}-\frac{419}{41}a-\frac{54}{41}$, $\frac{3}{41}a^{14}-\frac{7}{41}a^{13}-\frac{311}{164}a^{12}+\frac{365}{82}a^{11}+\frac{774}{41}a^{10}-\frac{3695}{82}a^{9}-\frac{3631}{41}a^{8}+\frac{441}{2}a^{7}+\frac{15233}{82}a^{6}-\frac{42735}{82}a^{5}-\frac{4009}{41}a^{4}+\frac{20032}{41}a^{3}-\frac{4410}{41}a^{2}-\frac{1836}{41}a+\frac{418}{41}$, $\frac{3}{41}a^{14}+\frac{1}{82}a^{13}-\frac{329}{164}a^{12}-\frac{91}{164}a^{11}+\frac{882}{41}a^{10}+\frac{15}{2}a^{9}-\frac{9309}{82}a^{8}-\frac{3357}{82}a^{7}+\frac{25027}{82}a^{6}+\frac{7099}{82}a^{5}-\frac{16099}{41}a^{4}-\frac{1660}{41}a^{3}+\frac{7748}{41}a^{2}-\frac{1239}{41}a-\frac{259}{41}$, $\frac{5}{164}a^{14}-\frac{4}{41}a^{13}-\frac{57}{82}a^{12}+\frac{11}{4}a^{11}+\frac{957}{164}a^{10}-\frac{1224}{41}a^{9}-\frac{1717}{82}a^{8}+\frac{12625}{82}a^{7}+\frac{1577}{82}a^{6}-\frac{15230}{41}a^{5}+\frac{2635}{41}a^{4}+\frac{13673}{41}a^{3}-\frac{5143}{41}a^{2}-\frac{90}{41}a+\frac{87}{41}$, $\frac{5}{164}a^{13}+\frac{19}{82}a^{12}-\frac{89}{164}a^{11}-\frac{871}{164}a^{10}+\frac{120}{41}a^{9}+\frac{1876}{41}a^{8}-\frac{167}{41}a^{7}-\frac{15205}{82}a^{6}+\frac{133}{82}a^{5}+\frac{14795}{41}a^{4}-\frac{1415}{41}a^{3}-\frac{11367}{41}a^{2}+\frac{2867}{41}a+\frac{345}{41}$, $\frac{7}{82}a^{14}-\frac{3}{41}a^{13}-\frac{88}{41}a^{12}+\frac{9}{4}a^{11}+\frac{3393}{164}a^{10}-\frac{1082}{41}a^{9}-\frac{7735}{82}a^{8}+\frac{11939}{82}a^{7}+\frac{7997}{41}a^{6}-\frac{30553}{82}a^{5}-\frac{4389}{41}a^{4}+\frac{14529}{41}a^{3}-\frac{3970}{41}a^{2}-\frac{170}{41}a-\frac{27}{41}$, $\frac{7}{82}a^{13}-\frac{33}{164}a^{12}-\frac{91}{41}a^{11}+\frac{355}{82}a^{10}+\frac{910}{41}a^{9}-\frac{1422}{41}a^{8}-\frac{4297}{41}a^{7}+\frac{5322}{41}a^{6}+\frac{18601}{82}a^{5}-\frac{9660}{41}a^{4}-\frac{6914}{41}a^{3}+\frac{7147}{41}a^{2}-\frac{1017}{41}a-\frac{141}{41}$, $\frac{7}{82}a^{14}-\frac{3}{41}a^{13}-\frac{393}{164}a^{12}+\frac{7}{4}a^{11}+\frac{4377}{164}a^{10}-\frac{672}{41}a^{9}-\frac{12245}{82}a^{8}+\frac{3161}{41}a^{7}+\frac{17960}{41}a^{6}-\frac{7917}{41}a^{5}-\frac{26119}{41}a^{4}+\frac{9814}{41}a^{3}+\frac{15013}{41}a^{2}-\frac{4475}{41}a-\frac{642}{41}$, $\frac{21}{164}a^{14}-\frac{47}{164}a^{13}-\frac{527}{164}a^{12}+\frac{1123}{164}a^{11}+\frac{5045}{164}a^{10}-\frac{5171}{82}a^{9}-\frac{5610}{41}a^{8}+\frac{11487}{41}a^{7}+\frac{10744}{41}a^{6}-\frac{49475}{82}a^{5}-\frac{3773}{41}a^{4}+\frac{20591}{41}a^{3}-\frac{7701}{41}a^{2}+\frac{197}{41}a+\frac{152}{41}$, $\frac{7}{164}a^{14}-\frac{3}{41}a^{13}-\frac{39}{41}a^{12}+\frac{279}{164}a^{11}+\frac{299}{41}a^{10}-\frac{1247}{82}a^{9}-\frac{1445}{82}a^{8}+\frac{5333}{82}a^{7}-\frac{1577}{41}a^{6}-\frac{5297}{41}a^{5}+\frac{9756}{41}a^{4}+\frac{2874}{41}a^{3}-\frac{11649}{41}a^{2}+\frac{2682}{41}a+\frac{413}{41}$, $\frac{21}{164}a^{14}-\frac{33}{164}a^{13}-\frac{140}{41}a^{12}+\frac{841}{164}a^{11}+\frac{5673}{164}a^{10}-\frac{4171}{82}a^{9}-\frac{6622}{41}a^{8}+\frac{10060}{41}a^{7}+\frac{13073}{41}a^{6}-\frac{23514}{41}a^{5}-\frac{4003}{41}a^{4}+\frac{20688}{41}a^{3}-\frac{11214}{41}a^{2}+28a+\frac{257}{41}$
| 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: | \( 22293487.2915 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 22293487.2915 \cdot 1}{2\cdot\sqrt{1730160900125000000000000}}\cr\approx \mathstrut & 0.277686458464 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 30*x^13 + 360*x^11 - 12*x^10 - 2200*x^9 + 240*x^8 + 7200*x^7 - 1680*x^6 - 11980*x^5 + 4800*x^4 + 7800*x^3 - 4800*x^2 + 400*x + 104)
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 - 30*x^13 + 360*x^11 - 12*x^10 - 2200*x^9 + 240*x^8 + 7200*x^7 - 1680*x^6 - 11980*x^5 + 4800*x^4 + 7800*x^3 - 4800*x^2 + 400*x + 104, 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 - 30*x^13 + 360*x^11 - 12*x^10 - 2200*x^9 + 240*x^8 + 7200*x^7 - 1680*x^6 - 11980*x^5 + 4800*x^4 + 7800*x^3 - 4800*x^2 + 400*x + 104);
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 - 30*x^13 + 360*x^11 - 12*x^10 - 2200*x^9 + 240*x^8 + 7200*x^7 - 1680*x^6 - 11980*x^5 + 4800*x^4 + 7800*x^3 - 4800*x^2 + 400*x + 104);
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
$C_3\times F_5$ (as 15T8):
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 60 |
| The 15 conjugacy class representatives for $F_5\times C_3$ |
| Character table for $F_5\times C_3$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 5.5.2450000.1 |
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
| Degree 30 sibling: | 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 | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | R | R | $15$ | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $15$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | $15$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.15.12.1 | $x^{15} + 5 x^{13} + 5 x^{12} + 10 x^{11} + 26 x^{10} + 20 x^{9} - 145 x^{7} + 70 x^{6} + 73 x^{5} + 315 x^{4} - 105 x^{3} + 200 x^{2} + 5 x + 1$ | $5$ | $3$ | $12$ | $F_5\times C_3$ | $[\ ]_{5}^{12}$ |
|
\(5\)
| 5.15.15.18 | $x^{15} - 60 x^{12} + 60 x^{11} + 15 x^{10} + 4425 x^{9} - 2400 x^{8} + 600 x^{7} + 17725 x^{6} + 88575 x^{5} - 1875 x^{4} - 4000 x^{3} + 4500 x^{2} + 1500 x + 125$ | $5$ | $3$ | $15$ | $F_5\times C_3$ | $[5/4]_{4}^{3}$ |
|
\(7\)
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.12.10.5 | $x^{12} - 154 x^{6} - 1421$ | $6$ | $2$ | $10$ | $C_{12}$ | $[\ ]_{6}^{2}$ |