Normalized defining polynomial
\( x^{15} - 2 x^{14} + x^{13} - x^{12} - 5 x^{11} + 4 x^{10} - 3 x^{9} + 2 x^{8} + 6 x^{7} - 3 x^{6} + \cdots - 1 \)
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: | \(-3703260525677583\) \(\medspace = -\,3\cdot 13^{3}\cdot 561866260913\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.91\) | 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: | $3^{1/2}13^{3/4}561866260913^{1/2}\approx 8888625.705547903$ | ||
Ramified primes: | \(3\), \(13\), \(561866260913\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-21912784175607}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{659}a^{14}+\frac{147}{659}a^{13}+\frac{157}{659}a^{12}+\frac{327}{659}a^{11}-\frac{48}{659}a^{10}+\frac{101}{659}a^{9}-\frac{111}{659}a^{8}-\frac{62}{659}a^{7}-\frac{6}{659}a^{6}-\frac{238}{659}a^{5}+\frac{127}{659}a^{4}-\frac{192}{659}a^{3}-\frac{276}{659}a^{2}-\frac{269}{659}a+\frac{115}{659}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $a$, $\frac{334}{659}a^{14}-\frac{986}{659}a^{13}+\frac{1036}{659}a^{12}-\frac{835}{659}a^{11}-\frac{875}{659}a^{10}+\frac{2102}{659}a^{9}-\frac{2147}{659}a^{8}+\frac{1698}{659}a^{7}-\frac{27}{659}a^{6}-\frac{1071}{659}a^{5}+\frac{1560}{659}a^{4}-\frac{2182}{659}a^{3}+\frac{735}{659}a^{2}-\frac{881}{659}a-\frac{471}{659}$, $\frac{295}{659}a^{14}-\frac{129}{659}a^{13}-\frac{1133}{659}a^{12}+\frac{910}{659}a^{11}-\frac{1639}{659}a^{10}-\frac{1178}{659}a^{9}+\frac{3500}{659}a^{8}-\frac{1815}{659}a^{7}+\frac{2184}{659}a^{6}+\frac{1621}{659}a^{5}-\frac{3393}{659}a^{4}+\frac{693}{659}a^{3}-\frac{2340}{659}a^{2}-\frac{1593}{659}a+\frac{316}{659}$, $\frac{893}{659}a^{14}-\frac{2506}{659}a^{13}+\frac{2470}{659}a^{12}-\frac{1903}{659}a^{11}-\frac{3324}{659}a^{10}+\frac{6500}{659}a^{9}-\frac{6204}{659}a^{8}+\frac{4603}{659}a^{7}+\frac{2550}{659}a^{6}-\frac{5608}{659}a^{5}+\frac{5994}{659}a^{4}-\frac{6047}{659}a^{3}-\frac{661}{659}a^{2}-\frac{341}{659}a-\frac{1427}{659}$, $\frac{830}{659}a^{14}-\frac{2541}{659}a^{13}+\frac{3123}{659}a^{12}-\frac{2734}{659}a^{11}-\frac{2936}{659}a^{10}+\frac{7386}{659}a^{9}-\frac{8437}{659}a^{8}+\frac{5873}{659}a^{7}+\frac{2269}{659}a^{6}-\frac{6430}{659}a^{5}+\frac{7219}{659}a^{4}-\frac{6472}{659}a^{3}+\frac{252}{659}a^{2}-\frac{528}{659}a-\frac{1423}{659}$, $\frac{1058}{659}a^{14}-\frac{2634}{659}a^{13}+\frac{2015}{659}a^{12}-\frac{1327}{659}a^{11}-\frac{4654}{659}a^{10}+\frac{6031}{659}a^{9}-\frac{4749}{659}a^{8}+\frac{3599}{659}a^{7}+\frac{4196}{659}a^{6}-\frac{4679}{659}a^{5}+\frac{4543}{659}a^{4}-\frac{6095}{659}a^{3}-\frac{2048}{659}a^{2}-\frac{1891}{659}a-\frac{1563}{659}$, $\frac{759}{659}a^{14}-\frac{1775}{659}a^{13}+\frac{1202}{659}a^{12}-\frac{909}{659}a^{11}-\frac{3482}{659}a^{10}+\frac{4169}{659}a^{9}-\frac{2533}{659}a^{8}+\frac{1708}{659}a^{7}+\frac{4013}{659}a^{6}-\frac{3371}{659}a^{5}+\frac{1497}{659}a^{4}-\frac{3384}{659}a^{3}-\frac{2558}{659}a^{2}-\frac{1199}{659}a-\frac{1021}{659}$ | 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: | \( 39.9823344692 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 39.9823344692 \cdot 1}{2\cdot\sqrt{3703260525677583}}\cr\approx \mathstrut & 0.254000805254 \end{aligned}\]
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 | $15$ | R | ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | $15$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | $15$ | $15$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
3.9.0.1 | $x^{9} + 2 x^{3} + 2 x^{2} + x + 1$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.4.3.4 | $x^{4} + 91$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
\(561866260913\) | $\Q_{561866260913}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |