Normalized defining polynomial
\( x^{10} - x^{9} - 11x^{8} + 2x^{7} + 49x^{6} + 3x^{5} + 2x^{4} + 190x^{3} - 202x^{2} + 165x + 11 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(633490494003125\) \(\medspace = 5^{5}\cdot 11^{4}\cdot 61^{4}\) | 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: | \(30.21\) | 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: | $5^{1/2}11^{1/2}61^{1/2}\approx 57.92236183029832$ | ||
Ramified primes: | \(5\), \(11\), \(61\) | 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: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{469}a^{8}+\frac{13}{469}a^{7}+\frac{6}{469}a^{6}-\frac{183}{469}a^{5}-\frac{220}{469}a^{4}-\frac{12}{67}a^{3}-\frac{7}{67}a^{2}+\frac{232}{469}a-\frac{125}{469}$, $\frac{1}{16199729}a^{9}-\frac{16001}{16199729}a^{8}+\frac{274425}{16199729}a^{7}-\frac{336102}{1246133}a^{6}-\frac{4289444}{16199729}a^{5}-\frac{7458639}{16199729}a^{4}-\frac{527754}{2314247}a^{3}-\frac{2912209}{16199729}a^{2}-\frac{2330416}{16199729}a+\frac{2444486}{16199729}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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)
| |
Fundamental units: | $\frac{30240}{2314247}a^{9}-\frac{19912}{2314247}a^{8}-\frac{346824}{2314247}a^{7}-\frac{6003}{178019}a^{6}+\frac{1552233}{2314247}a^{5}+\frac{930606}{2314247}a^{4}+\frac{56973}{2314247}a^{3}+\frac{4263868}{2314247}a^{2}-\frac{4547576}{2314247}a+\frac{3447558}{2314247}$, $\frac{6748}{2314247}a^{9}+\frac{313795}{16199729}a^{8}-\frac{498000}{16199729}a^{7}-\frac{310570}{1246133}a^{6}-\frac{1157736}{16199729}a^{5}+\frac{16504498}{16199729}a^{4}+\frac{2913102}{2314247}a^{3}+\frac{3037111}{2314247}a^{2}+\frac{59353146}{16199729}a-\frac{11837505}{16199729}$, $\frac{32912}{2314247}a^{9}-\frac{193405}{16199729}a^{8}-\frac{2858742}{16199729}a^{7}+\frac{31939}{1246133}a^{6}+\frac{14597823}{16199729}a^{5}+\frac{1385201}{16199729}a^{4}-\frac{2137487}{2314247}a^{3}+\frac{5446835}{2314247}a^{2}-\frac{21247466}{16199729}a-\frac{646760}{16199729}$, $\frac{893570}{16199729}a^{9}-\frac{871932}{16199729}a^{8}-\frac{1457125}{2314247}a^{7}+\frac{36673}{178019}a^{6}+\frac{43516272}{16199729}a^{5}-\frac{3922957}{16199729}a^{4}+\frac{1542334}{2314247}a^{3}+\frac{156409147}{16199729}a^{2}-\frac{25729965}{2314247}a+\frac{153691938}{16199729}$, $\frac{3266771}{16199729}a^{9}-\frac{3436210}{16199729}a^{8}-\frac{5291218}{2314247}a^{7}+\frac{92318}{178019}a^{6}+\frac{173627334}{16199729}a^{5}+\frac{15003385}{16199729}a^{4}-\frac{6080301}{2314247}a^{3}+\frac{536929327}{16199729}a^{2}-\frac{1618445}{34541}a+\frac{348839704}{16199729}$ | 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: | \( 3059.3129912043337 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 3059.3129912043337 \cdot 1}{2\cdot\sqrt{633490494003125}}\cr\approx \mathstrut & 0.378881437835894 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 5.1.450241.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.180686308456189658142111984384765625.2 |
Degree 10 sibling: | 10.0.425072121476096875.1 |
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 | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.10.0.1}{10} }$ | R | ${\href{/padicField/7.2.0.1}{2} }^{5}$ | R | ${\href{/padicField/13.2.0.1}{2} }^{5}$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.2.0.1}{2} }^{5}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.10.5.1 | $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(61\) | $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.671.2t1.a.a | $1$ | $ 11 \cdot 61 $ | \(\Q(\sqrt{-671}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3355.2t1.a.a | $1$ | $ 5 \cdot 11 \cdot 61 $ | \(\Q(\sqrt{-3355}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 2.16775.10t3.b.b | $2$ | $ 5^{2} \cdot 11 \cdot 61 $ | 10.2.633490494003125.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |
* | 2.671.5t2.a.a | $2$ | $ 11 \cdot 61 $ | 5.1.450241.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.671.5t2.a.b | $2$ | $ 11 \cdot 61 $ | 5.1.450241.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.16775.10t3.b.a | $2$ | $ 5^{2} \cdot 11 \cdot 61 $ | 10.2.633490494003125.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |