Normalized defining polynomial
\( x^{13} - x^{12} + 3x^{11} - x^{10} + 3x^{9} + 6x^{8} - 6x^{7} + x^{6} + 6x^{5} - 8x^{4} - 9x^{3} + 2x^{2} + 4x + 1 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-105628246955776\) \(\medspace = -\,2^{8}\cdot 257\cdot 1605487703\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.99\) | 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: | $2^{2}257^{1/2}1605487703^{1/2}\approx 2569390.0900283707$ | ||
Ramified primes: | \(2\), \(257\), \(1605487703\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-412610339671}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{143}a^{12}+\frac{48}{143}a^{11}+\frac{67}{143}a^{10}-\frac{7}{143}a^{9}-\frac{54}{143}a^{8}-\frac{6}{13}a^{7}+\frac{49}{143}a^{6}-\frac{29}{143}a^{5}+\frac{15}{143}a^{4}+\frac{12}{143}a^{3}+\frac{7}{143}a^{2}+\frac{59}{143}a+\frac{35}{143}$
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: | $a$, $\frac{459}{143}a^{12}-\frac{562}{143}a^{11}+\frac{1581}{143}a^{10}-\frac{782}{143}a^{9}+\frac{1669}{143}a^{8}+\frac{249}{13}a^{7}-\frac{3392}{143}a^{6}+\frac{2419}{143}a^{5}+\frac{2166}{143}a^{4}-\frac{4216}{143}a^{3}-\frac{2078}{143}a^{2}+\frac{1055}{143}a+\frac{621}{143}$, $\frac{496}{143}a^{12}-\frac{645}{143}a^{11}+\frac{1629}{143}a^{10}-\frac{898}{143}a^{9}+\frac{1530}{143}a^{8}+\frac{248}{13}a^{7}-\frac{4153}{143}a^{6}+\frac{1632}{143}a^{5}+\frac{2721}{143}a^{4}-\frac{5202}{143}a^{3}-\frac{3106}{143}a^{2}+\frac{2237}{143}a+\frac{1344}{143}$, $\frac{887}{143}a^{12}-\frac{1325}{143}a^{11}+\frac{3230}{143}a^{10}-\frac{2348}{143}a^{9}+\frac{3439}{143}a^{8}+\frac{359}{13}a^{7}-\frac{7874}{143}a^{6}+\frac{4593}{143}a^{5}+\frac{3438}{143}a^{4}-\frac{9519}{143}a^{3}-\frac{3372}{143}a^{2}+\frac{3999}{143}a+\frac{1730}{143}$, $\frac{571}{143}a^{12}-\frac{906}{143}a^{11}+\frac{2078}{143}a^{10}-\frac{1709}{143}a^{9}+\frac{2199}{143}a^{8}+\frac{188}{13}a^{7}-\frac{5340}{143}a^{6}+\frac{2317}{143}a^{5}+\frac{2130}{143}a^{4}-\frac{6447}{143}a^{3}-\frac{2581}{143}a^{2}+\frac{3230}{143}a+\frac{1681}{143}$, $\frac{1008}{143}a^{12}-\frac{1523}{143}a^{11}+\frac{3901}{143}a^{10}-\frac{3052}{143}a^{9}+\frac{4913}{143}a^{8}+\frac{322}{13}a^{7}-\frac{7379}{143}a^{6}+\frac{5517}{143}a^{5}+\frac{3251}{143}a^{4}-\frac{9354}{143}a^{3}-\frac{3669}{143}a^{2}+\frac{3416}{143}a+\frac{1532}{143}$, $\frac{1459}{143}a^{12}-\frac{2040}{143}a^{11}+\frac{5089}{143}a^{10}-\frac{3349}{143}a^{9}+\frac{5298}{143}a^{8}+\frac{632}{13}a^{7}-\frac{12164}{143}a^{6}+\frac{6023}{143}a^{5}+\frac{6727}{143}a^{4}-\frac{14953}{143}a^{3}-\frac{7376}{143}a^{2}+\frac{6430}{143}a+\frac{3589}{143}$ | 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: | \( 94.5194105196 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{5}\cdot 94.5194105196 \cdot 1}{2\cdot\sqrt{105628246955776}}\cr\approx \mathstrut & 0.360238645902 \end{aligned}\]
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 | R | ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.13.0.1}{13} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.13.0.1}{13} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.13.0.1}{13} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.8.2 | $x^{4} + 2 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ |
2.9.0.1 | $x^{9} + x^{4} + 1$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
\(257\) | $\Q_{257}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
\(1605487703\) | $\Q_{1605487703}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1605487703}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1605487703}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1605487703}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |