Normalized defining polynomial
\( x^{13} - x^{12} - 2 x^{11} - 2 x^{10} + 3 x^{9} + 4 x^{8} + 8 x^{7} - 4 x^{6} - 13 x^{5} + 3 x^{4} + \cdots + 2 \)
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: | \(-509211272757248\) \(\medspace = -\,2^{14}\cdot 31079789597\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.53\) | 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: | $2^{223/96}31079789597^{1/2}\approx 882077.2661754743$ | ||
Ramified primes: | \(2\), \(31079789597\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-31079789597}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{268}a^{12}+\frac{3}{67}a^{11}+\frac{5}{67}a^{10}-\frac{5}{134}a^{9}-\frac{127}{268}a^{8}+\frac{95}{268}a^{7}-\frac{97}{268}a^{6}-\frac{59}{268}a^{5}-\frac{55}{134}a^{4}+\frac{47}{268}a^{3}-\frac{57}{268}a^{2}-\frac{77}{268}a-\frac{31}{134}$
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: | $\frac{22}{67}a^{12}-\frac{217}{268}a^{11}-\frac{49}{268}a^{10}-\frac{9}{268}a^{9}+\frac{549}{268}a^{8}+\frac{13}{67}a^{7}+\frac{509}{268}a^{6}-\frac{653}{134}a^{5}-\frac{635}{268}a^{4}+\frac{1255}{268}a^{3}-\frac{115}{67}a^{2}-\frac{679}{268}a+\frac{287}{134}$, $\frac{33}{67}a^{12}-\frac{79}{134}a^{11}-\frac{107}{268}a^{10}-\frac{62}{67}a^{9}+\frac{321}{268}a^{8}+\frac{39}{134}a^{7}+\frac{216}{67}a^{6}-\frac{753}{268}a^{5}-\frac{785}{268}a^{4}+\frac{489}{134}a^{3}-\frac{221}{268}a^{2}-\frac{315}{268}a+\frac{129}{134}$, $\frac{61}{134}a^{12}-\frac{36}{67}a^{11}-\frac{53}{134}a^{10}-\frac{141}{134}a^{9}+\frac{159}{134}a^{8}+\frac{33}{134}a^{7}+\frac{515}{134}a^{6}-\frac{158}{67}a^{5}-\frac{139}{67}a^{4}+\frac{187}{134}a^{3}-\frac{30}{67}a^{2}-\frac{275}{134}a+\frac{52}{67}$, $\frac{3}{268}a^{12}-\frac{31}{268}a^{11}+\frac{15}{67}a^{10}+\frac{37}{268}a^{9}+\frac{21}{268}a^{8}-\frac{117}{268}a^{7}-\frac{45}{134}a^{6}-\frac{61}{67}a^{5}+\frac{103}{134}a^{4}+\frac{171}{134}a^{3}-\frac{93}{67}a^{2}+\frac{439}{268}a+\frac{41}{134}$, $\frac{42}{67}a^{12}-\frac{195}{268}a^{11}-\frac{129}{134}a^{10}-\frac{407}{268}a^{9}+\frac{253}{134}a^{8}+\frac{275}{134}a^{7}+\frac{1593}{268}a^{6}-\frac{733}{268}a^{5}-\frac{399}{67}a^{4}+\frac{57}{268}a^{3}-\frac{263}{268}a^{2}-\frac{371}{134}a+\frac{9}{67}$, $\frac{87}{134}a^{12}-\frac{257}{268}a^{11}-\frac{71}{268}a^{10}-\frac{333}{268}a^{9}+\frac{481}{268}a^{8}-\frac{43}{134}a^{7}+\frac{1279}{268}a^{6}-\frac{577}{134}a^{5}-\frac{313}{268}a^{4}+\frac{1277}{268}a^{3}-\frac{235}{67}a^{2}-\frac{735}{268}a+\frac{167}{134}$, $\frac{9}{268}a^{12}+\frac{27}{67}a^{11}-\frac{111}{134}a^{10}-\frac{45}{134}a^{9}-\frac{205}{268}a^{8}+\frac{587}{268}a^{7}+\frac{199}{268}a^{6}+\frac{943}{268}a^{5}-\frac{348}{67}a^{4}-\frac{649}{268}a^{3}+\frac{425}{268}a^{2}-\frac{23}{268}a-\frac{11}{134}$ | 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: | \( 379.340811006 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{5}\cdot 379.340811006 \cdot 1}{2\cdot\sqrt{509211272757248}}\cr\approx \mathstrut & 0.658475918562 \end{aligned}\]
Galois group
A non-solvable group of order 6227020800 |
The 101 conjugacy class representatives for $S_{13}$ are not computed |
Character table for $S_{13}$ is not computed |
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 | ${\href{/padicField/3.13.0.1}{13} }$ | ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.13.0.1}{13} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.13.0.1}{13} }$ | ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.13.0.1}{13} }$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.13.0.1}{13} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
2.8.14.8 | $x^{8} + 2 x^{7} + 4 x^{3} + 6 x^{2} + 4 x + 2$ | $8$ | $1$ | $14$ | $C_2 \wr S_4$ | $[4/3, 4/3, 2, 7/3, 7/3, 5/2]_{3}^{2}$ | |
\(31079789597\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |