Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 35*x^13 + 455*x^11 - 70*x^10 - 2800*x^9 + 1225*x^8 + 8575*x^7 - 6475*x^6 - 11375*x^5 + 12250*x^4 + 3500*x^3 - 6125*x^2 + 875)
gp: K = bnfinit(y^15 - 35*y^13 + 455*y^11 - 70*y^10 - 2800*y^9 + 1225*y^8 + 8575*y^7 - 6475*y^6 - 11375*y^5 + 12250*y^4 + 3500*y^3 - 6125*y^2 + 875, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 35*x^13 + 455*x^11 - 70*x^10 - 2800*x^9 + 1225*x^8 + 8575*x^7 - 6475*x^6 - 11375*x^5 + 12250*x^4 + 3500*x^3 - 6125*x^2 + 875);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 35*x^13 + 455*x^11 - 70*x^10 - 2800*x^9 + 1225*x^8 + 8575*x^7 - 6475*x^6 - 11375*x^5 + 12250*x^4 + 3500*x^3 - 6125*x^2 + 875)
\( x^{15} - 35 x^{13} + 455 x^{11} - 70 x^{10} - 2800 x^{9} + 1225 x^{8} + 8575 x^{7} - 6475 x^{6} + \cdots + 875 \)
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: |
\(15088641971036407470703125\)
\(\medspace = 3^{6}\cdot 5^{15}\cdot 7^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(47.71\) | 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: | $3^{1/2}5^{23/20}7^{14/15}\approx 67.78510966855065$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{5}a^{7}$, $\frac{1}{5}a^{8}$, $\frac{1}{5}a^{9}$, $\frac{1}{25}a^{10}$, $\frac{1}{25}a^{11}$, $\frac{1}{25}a^{12}$, $\frac{1}{25}a^{13}$, $\frac{1}{25525}a^{14}+\frac{116}{25525}a^{13}+\frac{148}{25525}a^{12}-\frac{189}{25525}a^{11}-\frac{28}{25525}a^{10}-\frac{51}{5105}a^{9}-\frac{70}{1021}a^{8}+\frac{97}{1021}a^{7}-\frac{222}{5105}a^{6}-\frac{501}{5105}a^{5}+\frac{378}{1021}a^{4}+\frac{435}{1021}a^{3}-\frac{450}{1021}a^{2}-\frac{374}{1021}a-\frac{502}{1021}$
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: |
$\frac{1454}{5105}a^{14}+\frac{12226}{25525}a^{13}-\frac{235004}{25525}a^{12}-\frac{397954}{25525}a^{11}+\frac{2672597}{25525}a^{10}+\frac{814611}{5105}a^{9}-\frac{2770141}{5105}a^{8}-\frac{3055416}{5105}a^{7}+\frac{1524201}{1021}a^{6}+\frac{818361}{1021}a^{5}-\frac{2061871}{1021}a^{4}-\frac{128233}{1021}a^{3}+\frac{951356}{1021}a^{2}-\frac{51107}{1021}a-\frac{135258}{1021}$, $\frac{14506}{25525}a^{14}+\frac{19487}{25525}a^{13}-\frac{481166}{25525}a^{12}-\frac{645521}{25525}a^{11}+\frac{5721874}{25525}a^{10}+\frac{265748}{1021}a^{9}-\frac{6315573}{5105}a^{8}-\frac{974911}{1021}a^{7}+\frac{18231899}{5105}a^{6}+\frac{5493973}{5105}a^{5}-\frac{5079998}{1021}a^{4}+\frac{351554}{1021}a^{3}+\frac{2451995}{1021}a^{2}-\frac{299824}{1021}a-\frac{384136}{1021}$, $\frac{6587}{25525}a^{14}+\frac{8552}{25525}a^{13}-\frac{219694}{25525}a^{12}-\frac{285203}{25525}a^{11}+\frac{2635566}{25525}a^{10}+\frac{592152}{5105}a^{9}-\frac{2942554}{5105}a^{8}-\frac{441279}{1021}a^{7}+\frac{8569011}{5105}a^{6}+\frac{2574747}{5105}a^{5}-\frac{2408872}{1021}a^{4}+\frac{130086}{1021}a^{3}+\frac{1192341}{1021}a^{2}-\frac{125448}{1021}a-\frac{193645}{1021}$, $\frac{6587}{25525}a^{14}+\frac{8552}{25525}a^{13}-\frac{219694}{25525}a^{12}-\frac{285203}{25525}a^{11}+\frac{2635566}{25525}a^{10}+\frac{592152}{5105}a^{9}-\frac{2942554}{5105}a^{8}-\frac{441279}{1021}a^{7}+\frac{8569011}{5105}a^{6}+\frac{2574747}{5105}a^{5}-\frac{2408872}{1021}a^{4}+\frac{130086}{1021}a^{3}+\frac{1192341}{1021}a^{2}-\frac{125448}{1021}a-\frac{192624}{1021}$, $\frac{31713}{25525}a^{14}+\frac{9198}{5105}a^{13}-\frac{41739}{1021}a^{12}-\frac{1513609}{25525}a^{11}+\frac{489643}{1021}a^{10}+\frac{3108846}{5105}a^{9}-\frac{13265091}{5105}a^{8}-\frac{11492936}{5105}a^{7}+\frac{37785698}{5105}a^{6}+\frac{2766211}{1021}a^{5}-\frac{10446919}{1021}a^{4}+\frac{354711}{1021}a^{3}+\frac{4984189}{1021}a^{2}-\frac{517352}{1021}a-\frac{761139}{1021}$, $\frac{2523}{25525}a^{14}+\frac{1683}{25525}a^{13}-\frac{87067}{25525}a^{12}-\frac{57216}{25525}a^{11}+\frac{221518}{5105}a^{10}+\frac{107178}{5105}a^{9}-\frac{268500}{1021}a^{8}-\frac{225144}{5105}a^{7}+\frac{4227363}{5105}a^{6}-\frac{130693}{1021}a^{5}-\frac{1278212}{1021}a^{4}+\frac{463464}{1021}a^{3}+\frac{719807}{1021}a^{2}-\frac{211545}{1021}a-\frac{153656}{1021}$, $\frac{14}{1021}a^{14}+\frac{2823}{25525}a^{13}-\frac{5376}{25525}a^{12}-\frac{85549}{25525}a^{11}-\frac{57787}{25525}a^{10}+\frac{166951}{5105}a^{9}+\frac{237913}{5105}a^{8}-\frac{127368}{1021}a^{7}-\frac{1014978}{5105}a^{6}+\frac{241621}{1021}a^{5}+\frac{325269}{1021}a^{4}-\frac{232667}{1021}a^{3}-\frac{182004}{1021}a^{2}+\frac{67174}{1021}a+\frac{36668}{1021}$, $\frac{2436}{25525}a^{14}+\frac{2822}{25525}a^{13}-\frac{16313}{5105}a^{12}-\frac{94886}{25525}a^{11}+\frac{197297}{5105}a^{10}+\frac{196358}{5105}a^{9}-\frac{1114997}{5105}a^{8}-\frac{713516}{5105}a^{7}+\frac{3285916}{5105}a^{6}+\frac{738863}{5105}a^{5}-\frac{936391}{1021}a^{4}+\frac{87668}{1021}a^{3}+\frac{480224}{1021}a^{2}-\frac{60571}{1021}a-\frac{81394}{1021}$, $a-1$, $\frac{5752}{5105}a^{14}+\frac{7666}{5105}a^{13}-\frac{38229}{1021}a^{12}-\frac{1274044}{25525}a^{11}+\frac{11394649}{25525}a^{10}+\frac{2636639}{5105}a^{9}-\frac{12607269}{5105}a^{8}-\frac{9778419}{5105}a^{7}+\frac{36370676}{5105}a^{6}+\frac{2265109}{1021}a^{5}-\frac{10106186}{1021}a^{4}+\frac{586341}{1021}a^{3}+\frac{4850967}{1021}a^{2}-\frac{548282}{1021}a-\frac{745910}{1021}$, $\frac{54907}{25525}a^{14}+\frac{80873}{25525}a^{13}-\frac{1802989}{25525}a^{12}-\frac{2656621}{25525}a^{11}+\frac{21079796}{25525}a^{10}+\frac{5447381}{5105}a^{9}-\frac{22740921}{5105}a^{8}-\frac{20109443}{5105}a^{7}+\frac{64581678}{5105}a^{6}+\frac{24296112}{5105}a^{5}-\frac{17821597}{1021}a^{4}+\frac{540401}{1021}a^{3}+\frac{8466182}{1021}a^{2}-\frac{868716}{1021}a-\frac{1287879}{1021}$, $\frac{4539}{25525}a^{14}+\frac{5814}{25525}a^{13}-\frac{151154}{25525}a^{12}-\frac{7728}{1021}a^{11}+\frac{361949}{5105}a^{10}+\frac{397447}{5105}a^{9}-\frac{2016449}{5105}a^{8}-\frac{1445597}{5105}a^{7}+\frac{5865714}{5105}a^{6}+\frac{1552669}{5105}a^{5}-\frac{1643348}{1021}a^{4}+\frac{140749}{1021}a^{3}+\frac{803998}{1021}a^{2}-\frac{102784}{1021}a-\frac{129373}{1021}$, $\frac{6471}{25525}a^{14}+\frac{10411}{25525}a^{13}-\frac{210316}{25525}a^{12}-\frac{339854}{25525}a^{11}+\frac{483043}{5105}a^{10}+\frac{696084}{5105}a^{9}-\frac{2539499}{5105}a^{8}-\frac{519917}{1021}a^{7}+\frac{7064284}{5105}a^{6}+\frac{3393508}{5105}a^{5}-\frac{1918737}{1021}a^{4}-\frac{65356}{1021}a^{3}+\frac{883107}{1021}a^{2}-\frac{53476}{1021}a-\frac{124182}{1021}$, $\frac{9333}{25525}a^{14}+\frac{2524}{5105}a^{13}-\frac{309492}{25525}a^{12}-\frac{418259}{25525}a^{11}+\frac{3677694}{25525}a^{10}+\frac{862548}{5105}a^{9}-\frac{810544}{1021}a^{8}-\frac{3181024}{5105}a^{7}+\frac{11667671}{5105}a^{6}+\frac{3655527}{5105}a^{5}-\frac{3240335}{1021}a^{4}+\frac{194349}{1021}a^{3}+\frac{1555527}{1021}a^{2}-\frac{179439}{1021}a-\frac{240753}{1021}$
| 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: | \( 63554489.595 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 63554489.595 \cdot 1}{2\cdot\sqrt{15088641971036407470703125}}\cr\approx \mathstrut & 0.26806560731 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 35*x^13 + 455*x^11 - 70*x^10 - 2800*x^9 + 1225*x^8 + 8575*x^7 - 6475*x^6 - 11375*x^5 + 12250*x^4 + 3500*x^3 - 6125*x^2 + 875)
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 - 35*x^13 + 455*x^11 - 70*x^10 - 2800*x^9 + 1225*x^8 + 8575*x^7 - 6475*x^6 - 11375*x^5 + 12250*x^4 + 3500*x^3 - 6125*x^2 + 875, 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 - 35*x^13 + 455*x^11 - 70*x^10 - 2800*x^9 + 1225*x^8 + 8575*x^7 - 6475*x^6 - 11375*x^5 + 12250*x^4 + 3500*x^3 - 6125*x^2 + 875);
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 - 35*x^13 + 455*x^11 - 70*x^10 - 2800*x^9 + 1225*x^8 + 8575*x^7 - 6475*x^6 - 11375*x^5 + 12250*x^4 + 3500*x^3 - 6125*x^2 + 875);
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
| \(\Q(\zeta_{7})^+\), 5.5.67528125.1 |
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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | R | R | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | $15$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 3.12.6.1 | $x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ | |
|
\(5\)
| 5.15.15.18 | $x^{15} - 60 x^{12} + 60 x^{11} + 15 x^{10} + 4425 x^{9} - 2400 x^{8} + 600 x^{7} + 17725 x^{6} + 88575 x^{5} - 1875 x^{4} - 4000 x^{3} + 4500 x^{2} + 1500 x + 125$ | $5$ | $3$ | $15$ | $F_5\times C_3$ | $[5/4]_{4}^{3}$ |
|
\(7\)
| 7.15.14.1 | $x^{15} + 7$ | $15$ | $1$ | $14$ | $F_5\times C_3$ | $[\ ]_{15}^{4}$ |