Normalized defining polynomial
\( x^{12} - 2 x^{11} - 4 x^{10} + 10 x^{9} + 15 x^{8} - 38 x^{7} - 24 x^{6} + 72 x^{5} + 30 x^{4} + \cdots + 31 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1600000000000000\) \(\medspace = 2^{18}\cdot 5^{14}\) | 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: | \(18.49\) | 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: | $2^{3/2}5^{13/10}\approx 22.91954538992328$ | ||
Ramified primes: | \(2\), \(5\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{23}a^{10}+\frac{11}{23}a^{9}-\frac{2}{23}a^{8}-\frac{3}{23}a^{7}+\frac{5}{23}a^{6}-\frac{10}{23}a^{5}-\frac{8}{23}a^{4}-\frac{2}{23}a^{3}+\frac{5}{23}a^{2}-\frac{9}{23}a-\frac{5}{23}$, $\frac{1}{20953}a^{11}-\frac{186}{20953}a^{10}-\frac{398}{20953}a^{9}+\frac{293}{911}a^{8}+\frac{2643}{20953}a^{7}-\frac{5342}{20953}a^{6}-\frac{7353}{20953}a^{5}+\frac{2011}{20953}a^{4}-\frac{9238}{20953}a^{3}+\frac{8896}{20953}a^{2}-\frac{8030}{20953}a+\frac{8115}{20953}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $5$ | 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: | $\frac{2608}{20953}a^{11}-\frac{3169}{20953}a^{10}-\frac{11287}{20953}a^{9}+\frac{726}{911}a^{8}+\frac{41313}{20953}a^{7}-\frac{61050}{20953}a^{6}-\frac{67488}{20953}a^{5}+\frac{111203}{20953}a^{4}+\frac{66105}{20953}a^{3}-\frac{98968}{20953}a^{2}-\frac{31146}{20953}a+\frac{1390}{20953}$, $\frac{577}{20953}a^{11}-\frac{2557}{20953}a^{10}+\frac{837}{20953}a^{9}+\frac{526}{911}a^{8}-\frac{4558}{20953}a^{7}-\frac{44149}{20953}a^{6}+\frac{31731}{20953}a^{5}+\frac{70791}{20953}a^{4}-\frac{50170}{20953}a^{3}-\frac{63352}{20953}a^{2}+\frac{18256}{20953}a+\frac{30789}{20953}$, $\frac{141}{20953}a^{11}-\frac{1629}{20953}a^{10}+\frac{4919}{20953}a^{9}+\frac{26}{20953}a^{8}-\frac{15423}{20953}a^{7}-\frac{1647}{20953}a^{6}+\frac{58249}{20953}a^{5}-\frac{38943}{20953}a^{4}-\frac{52666}{20953}a^{3}+\frac{36329}{20953}a^{2}+\frac{29295}{20953}a-\frac{5467}{20953}$, $\frac{386}{20953}a^{11}-\frac{2560}{20953}a^{10}+\frac{331}{20953}a^{9}+\frac{11281}{20953}a^{8}-\frac{4677}{20953}a^{7}-\frac{39592}{20953}a^{6}+\frac{31392}{20953}a^{5}+\frac{54734}{20953}a^{4}-\frac{58518}{20953}a^{3}-\frac{33410}{20953}a^{2}+\frac{48836}{20953}a-\frac{539}{20953}$, $\frac{1423}{20953}a^{11}-\frac{2310}{20953}a^{10}-\frac{6089}{20953}a^{9}+\frac{13165}{20953}a^{8}+\frac{19512}{20953}a^{7}-\frac{45832}{20953}a^{6}-\frac{33280}{20953}a^{5}+\frac{92213}{20953}a^{4}+\frac{11899}{20953}a^{3}-\frac{67662}{20953}a^{2}-\frac{42834}{20953}a+\frac{467}{911}$ | 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: | \( 534.541702856 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 534.541702856 \cdot 2}{2\cdot\sqrt{1600000000000000}}\cr\approx \mathstrut & 0.822244186626 \end{aligned}\]
Galois group
A non-solvable group of order 60 |
The 5 conjugacy class representatives for $A_5$ |
Character table for $A_5$ |
Intermediate fields
6.2.1000000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.1000000.2 |
Degree 6 sibling: | 6.2.1000000.1 |
Degree 10 sibling: | 10.2.1000000000000.1 |
Degree 15 sibling: | deg 15 |
Degree 20 sibling: | deg 20 |
Degree 30 sibling: | data not computed |
Minimal sibling: | 5.1.1000000.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.18.59 | $x^{12} + 6 x^{11} + 22 x^{10} + 56 x^{9} + 126 x^{8} + 240 x^{7} + 332 x^{6} - 18 x^{5} - 459 x^{4} - 394 x^{3} - 344 x^{2} + 138 x + 423$ | $4$ | $3$ | $18$ | $A_4$ | $[2, 2]^{3}$ |
\(5\) | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.10.13.7 | $x^{10} + 5 x^{4} + 10$ | $10$ | $1$ | $13$ | $D_5$ | $[3/2]_{2}$ |