Normalized defining polynomial
\( x^{15} - 2 x^{14} - 5 x^{13} + 10 x^{12} + 3 x^{11} - 12 x^{10} + 14 x^{9} - 6 x^{8} - 11 x^{7} + \cdots - 1 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[9, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(-1661733554869524611658797104\)
\(\medspace = -\,2^{4}\cdot 8009\cdot 12967704729597363994091\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(65.27\) | 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: | $2^{4/5}8009^{1/2}12967704729597363994091^{1/2}\approx 17743720995897.23$ | ||
Ramified primes: |
\(2\), \(8009\), \(12967704729597363994091\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-10385\!\cdots\!74819}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | 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: |
$a$, $a^{14}-2a^{13}-5a^{12}+10a^{11}+3a^{10}-12a^{9}+14a^{8}-6a^{7}-11a^{6}+20a^{5}-12a^{4}-21a^{3}+11a^{2}+9a$, $a^{14}-a^{13}-6a^{12}+4a^{11}+7a^{10}-5a^{9}+9a^{8}+3a^{7}-8a^{6}+12a^{5}-21a^{3}-10a^{2}$, $a^{14}-2a^{13}-5a^{12}+10a^{11}+3a^{10}-12a^{9}+14a^{8}-6a^{7}-11a^{6}+20a^{5}-12a^{4}-21a^{3}+10a^{2}+9a+1$, $3a^{14}-4a^{13}-15a^{12}+13a^{11}+10a^{10}+25a^{8}-13a^{7}+7a^{6}+9a^{5}-7a^{4}-37a^{3}-51a^{2}-2a+9$, $4a^{14}-8a^{13}-19a^{12}+38a^{11}+7a^{10}-37a^{9}+56a^{8}-36a^{7}-24a^{6}+66a^{5}-53a^{4}-62a^{3}+24a^{2}+29a+6$, $7a^{14}-18a^{13}-24a^{12}+82a^{11}-28a^{10}-61a^{9}+129a^{8}-119a^{7}+2a^{6}+127a^{5}-151a^{4}-54a^{3}+95a^{2}+16a-12$, $20a^{14}-48a^{13}-83a^{12}+237a^{11}-24a^{10}-247a^{9}+373a^{8}-255a^{7}-145a^{6}+473a^{5}-421a^{4}-284a^{3}+361a^{2}+89a-59$, $7a^{14}-18a^{13}-25a^{12}+85a^{11}-26a^{10}-73a^{9}+137a^{8}-115a^{7}-10a^{6}+146a^{5}-166a^{4}-53a^{3}+105a^{2}+9a-12$, $7a^{14}-17a^{13}-28a^{12}+82a^{11}-14a^{10}-70a^{9}+125a^{8}-119a^{7}-19a^{6}+147a^{5}-135a^{4}-53a^{3}+88a^{2}+9a-10$, $a^{14}+a^{13}-9a^{12}-4a^{11}+17a^{10}-9a^{9}+7a^{8}+28a^{7}-23a^{6}+30a^{5}+21a^{4}-26a^{3}-8a^{2}-4a-3$
| 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: | \( 375432035.765 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{3}\cdot 375432035.765 \cdot 1}{2\cdot\sqrt{1661733554869524611658797104}}\cr\approx \mathstrut & 0.584830756131 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 1307674368000 |
The 176 conjugacy class representatives for $S_{15}$ |
Character table for $S_{15}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.5.0.1}{5} }$ | ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $15$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\)
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
2.5.4.1 | $x^{5} + 2$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
\(8009\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(129\!\cdots\!091\)
| $\Q_{12\!\cdots\!91}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |