Normalized defining polynomial
\( x^{10} - 3x^{9} - 2x^{8} + 24x^{7} - 28x^{6} - 21x^{5} + 47x^{4} - 2x^{3} - 28x^{2} - 8x + 16 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-13636502136971\) \(\medspace = -\,7^{3}\cdot 3413^{3}\) | 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: | \(20.58\) | 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: | $7^{1/2}3413^{1/2}\approx 154.56713751635564$ | ||
Ramified primes: | \(7\), \(3413\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{-23891}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8488}a^{9}+\frac{31}{1061}a^{8}+\frac{177}{2122}a^{7}+\frac{803}{4244}a^{6}+\frac{2071}{4244}a^{5}+\frac{4085}{8488}a^{4}-\frac{833}{4244}a^{3}-\frac{67}{4244}a^{2}-\frac{229}{1061}a-\frac{186}{1061}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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)
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Fundamental units: | $\frac{351}{4244}a^{9}-\frac{969}{8488}a^{8}-\frac{472}{1061}a^{7}+\frac{6681}{4244}a^{6}+\frac{1333}{4244}a^{5}-\frac{7215}{2122}a^{4}+\frac{13483}{8488}a^{3}+\frac{11321}{4244}a^{2}-\frac{1608}{1061}a-\frac{1130}{1061}$, $\frac{221}{8488}a^{9}-\frac{91}{2122}a^{8}-\frac{70}{1061}a^{7}+\frac{1337}{4244}a^{6}-\frac{661}{4244}a^{5}+\frac{3057}{8488}a^{4}-\frac{3723}{4244}a^{3}+\frac{47}{4244}a^{2}-\frac{423}{2122}a+\frac{273}{1061}$, $\frac{33}{8488}a^{9}-\frac{38}{1061}a^{8}+\frac{11}{4244}a^{7}+\frac{1035}{4244}a^{6}-\frac{1683}{4244}a^{5}-\frac{5247}{8488}a^{4}+\frac{4341}{4244}a^{3}+\frac{1547}{2122}a^{2}-\frac{1321}{2122}a-\frac{833}{1061}$, $\frac{179}{8488}a^{9}-\frac{85}{4244}a^{8}-\frac{147}{2122}a^{7}+\frac{1563}{4244}a^{6}-\frac{641}{4244}a^{5}-\frac{2997}{8488}a^{4}+\frac{1715}{1061}a^{3}-\frac{1383}{4244}a^{2}-\frac{1734}{1061}a-\frac{1464}{1061}$, $\frac{247}{8488}a^{9}-\frac{1343}{8488}a^{8}+\frac{109}{1061}a^{7}+\frac{2089}{2122}a^{6}-\frac{4707}{2122}a^{5}+\frac{1045}{8488}a^{4}+\frac{26691}{8488}a^{3}-\frac{317}{2122}a^{2}-\frac{330}{1061}a-\frac{319}{1061}$, $\frac{1489}{8488}a^{9}-\frac{525}{1061}a^{8}-\frac{318}{1061}a^{7}+\frac{15835}{4244}a^{6}-\frac{18645}{4244}a^{5}-\frac{11819}{8488}a^{4}+\frac{15887}{4244}a^{3}-\frac{29}{4244}a^{2}-\frac{1461}{1061}a-\frac{2155}{1061}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 627.63937747 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 627.63937747 \cdot 1}{2\cdot\sqrt{13636502136971}}\cr\approx \mathstrut & 0.33727820867 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 5 sibling: | 5.3.23891.1 |
Degree 6 sibling: | 6.0.13636502136971.1 |
Degree 10 sibling: | data not computed |
Degree 12 sibling: | data not computed |
Degree 15 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 sibling: | data not computed |
Minimal sibling: | 5.3.23891.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(3413\) | $\Q_{3413}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $2$ | $3$ | $3$ |