Normalized defining polynomial
\( x^{13} - x^{12} + 2x^{11} + 2x^{10} - 3x^{9} + x^{8} - x^{7} - 8x^{6} - 7x^{5} - 6x^{4} - 6x^{3} - 6x^{2} - 4x - 1 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-6855574490939\) \(\medspace = -\,1798333\cdot 3812183\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(9.71\) | 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))
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Galois root discriminant: | $1798333^{1/2}3812183^{1/2}\approx 2618315.200837936$ | ||
Ramified primes: | \(1798333\), \(3812183\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-6855574490939}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$
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)
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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: | $\frac{23}{2}a^{12}-18a^{11}+\frac{67}{2}a^{10}+\frac{7}{2}a^{9}-36a^{8}+32a^{7}-\frac{61}{2}a^{6}-75a^{5}-38a^{4}-\frac{95}{2}a^{3}-43a^{2}-45a-\frac{39}{2}$, $\frac{9}{2}a^{12}-\frac{15}{2}a^{11}+14a^{10}-\frac{1}{2}a^{9}-\frac{27}{2}a^{8}+\frac{27}{2}a^{7}-14a^{6}-27a^{5}-12a^{4}-\frac{35}{2}a^{3}-\frac{27}{2}a^{2}-\frac{31}{2}a-6$, $a^{12}-a^{11}+2a^{10}+2a^{9}-3a^{8}+a^{7}-a^{6}-8a^{5}-7a^{4}-6a^{3}-6a^{2}-6a-4$, $\frac{7}{2}a^{12}-5a^{11}+9a^{10}+\frac{7}{2}a^{9}-13a^{8}+\frac{19}{2}a^{7}-\frac{15}{2}a^{6}-\frac{51}{2}a^{5}-\frac{25}{2}a^{4}-14a^{3}-\frac{27}{2}a^{2}-14a-7$, $8a^{12}-\frac{25}{2}a^{11}+23a^{10}+3a^{9}-\frac{51}{2}a^{8}+22a^{7}-\frac{41}{2}a^{6}-\frac{103}{2}a^{5}-\frac{55}{2}a^{4}-\frac{65}{2}a^{3}-28a^{2}-\frac{65}{2}a-14$, $5a^{12}-\frac{17}{2}a^{11}+16a^{10}-a^{9}-\frac{29}{2}a^{8}+16a^{7}-\frac{33}{2}a^{6}-\frac{59}{2}a^{5}-\frac{27}{2}a^{4}-\frac{45}{2}a^{3}-17a^{2}-\frac{35}{2}a-8$, $6a^{12}-\frac{19}{2}a^{11}+\frac{35}{2}a^{10}+2a^{9}-\frac{39}{2}a^{8}+\frac{35}{2}a^{7}-\frac{31}{2}a^{6}-40a^{5}-18a^{4}-25a^{3}-\frac{45}{2}a^{2}-\frac{47}{2}a-\frac{21}{2}$ | 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: | \( 10.9027200388 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{5}\cdot 10.9027200388 \cdot 1}{2\cdot\sqrt{6855574490939}}\cr\approx \mathstrut & 0.163106874760 \end{aligned}\]
Galois group
A non-solvable group of order 6227020800 |
The 101 conjugacy class representatives for $S_{13}$ |
Character table for $S_{13}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 26 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 | ${\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.13.0.1}{13} }$ | ${\href{/padicField/5.13.0.1}{13} }$ | ${\href{/padicField/7.13.0.1}{13} }$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.5.0.1}{5} }$ | ${\href{/padicField/29.13.0.1}{13} }$ | ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.13.0.1}{13} }$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(1798333\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
\(3812183\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |