Normalized defining polynomial
\( x^{10} - x^{9} - 10x^{8} + 22x^{7} + 69x^{6} - 221x^{5} + 208x^{4} - 32x^{3} + 902x^{2} - 2018x + 2552 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-26026835840008192\) \(\medspace = -\,2^{16}\cdot 7^{5}\cdot 4861^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(43.81\) | 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: | $2^{13/6}7^{1/2}4861^{1/2}\approx 828.2157551244516$ | ||
Ramified primes: | \(2\), \(7\), \(4861\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | unavailable$^{16}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{748147492706}a^{9}+\frac{34862003743}{748147492706}a^{8}+\frac{134178560092}{374073746353}a^{7}-\frac{159443384658}{374073746353}a^{6}-\frac{209821615519}{748147492706}a^{5}-\frac{219210023935}{748147492706}a^{4}-\frac{69865802806}{374073746353}a^{3}-\frac{170094900208}{374073746353}a^{2}+\frac{13841438499}{374073746353}a-\frac{170934807594}{374073746353}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{132}$, which has order $132$
Unit group
Rank: | $4$ | 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: | $\frac{256177578}{374073746353}a^{9}+\frac{86883817}{374073746353}a^{8}+\frac{2319216676}{374073746353}a^{7}-\frac{4006743104}{374073746353}a^{6}-\frac{18418804930}{374073746353}a^{5}+\frac{40165205078}{374073746353}a^{4}-\frac{47475574184}{374073746353}a^{3}+\frac{14295986520}{374073746353}a^{2}-\frac{27726038955}{374073746353}a-\frac{177065205287}{374073746353}$, $\frac{284889822}{374073746353}a^{9}+\frac{1311131366}{374073746353}a^{8}-\frac{4758327892}{374073746353}a^{7}-\frac{20451159392}{374073746353}a^{6}+\frac{52555735528}{374073746353}a^{5}+\frac{125340554405}{374073746353}a^{4}-\frac{392605163310}{374073746353}a^{3}-\frac{157057481938}{374073746353}a^{2}+\frac{1161578791831}{374073746353}a-\frac{1889187466209}{374073746353}$, $\frac{338587522}{374073746353}a^{9}-\frac{485118208}{374073746353}a^{8}+\frac{3258327224}{374073746353}a^{7}-\frac{1485104096}{374073746353}a^{6}-\frac{32900079320}{374073746353}a^{5}+\frac{11618724080}{374073746353}a^{4}+\frac{12196608048}{374073746353}a^{3}+\frac{32618526449}{374073746353}a^{2}-\frac{265841876323}{374073746353}a+\frac{185309119053}{374073746353}$, $\frac{82409944}{374073746353}a^{9}-\frac{572002025}{374073746353}a^{8}+\frac{939110548}{374073746353}a^{7}+\frac{2521639008}{374073746353}a^{6}-\frac{14481274390}{374073746353}a^{5}-\frac{28546480998}{374073746353}a^{4}+\frac{59672182232}{374073746353}a^{3}+\frac{18322539929}{374073746353}a^{2}-\frac{238115837368}{374073746353}a-\frac{759846914719}{374073746353}$ | 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: | \( 999.3743297020859 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 999.3743297020859 \cdot 132}{2\cdot\sqrt{26026835840008192}}\cr\approx \mathstrut & 4.00369311047783 \end{aligned}\]
Galois group
$C_2\times S_5$ (as 10T22):
A non-solvable group of order 240 |
The 14 conjugacy class representatives for $S_5\times C_2$ |
Character table for $S_5\times C_2$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), 5.5.1244416.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 10 sibling: | data not computed |
Degree 12 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 siblings: | 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 | ${\href{/padicField/3.10.0.1}{10} }$ | ${\href{/padicField/5.10.0.1}{10} }$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{5}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
2.4.8.8 | $x^{4} + 4 x + 2$ | $4$ | $1$ | $8$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
2.4.8.8 | $x^{4} + 4 x + 2$ | $4$ | $1$ | $8$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
\(7\) | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(4861\) | 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 $4$ | $2$ | $2$ | $2$ |