Normalized defining polynomial
\( x^{10} + 110 x^{8} - 10 x^{7} + 5840 x^{6} + 358 x^{5} + 180785 x^{4} + 39440 x^{3} + 3124530 x^{2} + \cdots + 23284457 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-1856465000000000000000\) \(\medspace = -\,2^{15}\cdot 5^{16}\cdot 13^{5}\) | 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: | \(133.93\) | 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: | $2^{3/2}5^{8/5}13^{1/2}\approx 133.9271652764882$ | ||
Ramified primes: | \(2\), \(5\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-26}) \) | ||
$\card{ \Gal(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(2600=2^{3}\cdot 5^{2}\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{2600}(1,·)$, $\chi_{2600}(2131,·)$, $\chi_{2600}(1091,·)$, $\chi_{2600}(2081,·)$, $\chi_{2600}(521,·)$, $\chi_{2600}(1611,·)$, $\chi_{2600}(1041,·)$, $\chi_{2600}(51,·)$, $\chi_{2600}(1561,·)$, $\chi_{2600}(571,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-26}) \), 10.0.1856465000000000000000.3$^{15}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{3}{7}a^{5}+\frac{2}{7}a^{4}+\frac{1}{7}a^{3}-\frac{3}{7}a^{2}+\frac{2}{7}a$, $\frac{1}{7}a^{7}-\frac{1}{7}a$, $\frac{1}{7}a^{8}-\frac{1}{7}a^{2}$, $\frac{1}{51\!\cdots\!43}a^{9}-\frac{23\!\cdots\!69}{11\!\cdots\!01}a^{8}-\frac{31\!\cdots\!59}{51\!\cdots\!43}a^{7}+\frac{35\!\cdots\!99}{51\!\cdots\!43}a^{6}+\frac{24\!\cdots\!51}{51\!\cdots\!43}a^{5}-\frac{15\!\cdots\!23}{51\!\cdots\!43}a^{4}+\frac{21\!\cdots\!78}{51\!\cdots\!43}a^{3}+\frac{43\!\cdots\!03}{51\!\cdots\!43}a^{2}-\frac{52\!\cdots\!03}{10\!\cdots\!07}a-\frac{11\!\cdots\!33}{24\!\cdots\!49}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $7$ |
Class group and class number
$C_{5}\times C_{8430}$, which has order $42150$ (assuming GRH)
Unit group
Rank: | $4$ | 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: | $\frac{80109426971100}{51\!\cdots\!43}a^{9}-\frac{39426865390484}{11\!\cdots\!01}a^{8}+\frac{81\!\cdots\!56}{51\!\cdots\!43}a^{7}-\frac{13\!\cdots\!90}{51\!\cdots\!43}a^{6}+\frac{36\!\cdots\!84}{51\!\cdots\!43}a^{5}-\frac{53\!\cdots\!67}{51\!\cdots\!43}a^{4}+\frac{65\!\cdots\!40}{51\!\cdots\!43}a^{3}-\frac{11\!\cdots\!48}{51\!\cdots\!43}a^{2}+\frac{66\!\cdots\!60}{10\!\cdots\!07}a-\frac{48\!\cdots\!98}{24\!\cdots\!49}$, $\frac{51303764447020}{51\!\cdots\!43}a^{9}+\frac{30337540818901}{11\!\cdots\!01}a^{8}+\frac{53\!\cdots\!28}{51\!\cdots\!43}a^{7}+\frac{11\!\cdots\!86}{51\!\cdots\!43}a^{6}+\frac{22\!\cdots\!76}{51\!\cdots\!43}a^{5}+\frac{51\!\cdots\!28}{51\!\cdots\!43}a^{4}+\frac{68\!\cdots\!56}{51\!\cdots\!43}a^{3}+\frac{12\!\cdots\!02}{51\!\cdots\!43}a^{2}+\frac{19\!\cdots\!60}{10\!\cdots\!07}a+\frac{65\!\cdots\!02}{24\!\cdots\!49}$, $\frac{104555056314090}{51\!\cdots\!43}a^{9}+\frac{56576385521422}{11\!\cdots\!01}a^{8}+\frac{10\!\cdots\!46}{51\!\cdots\!43}a^{7}+\frac{20\!\cdots\!56}{51\!\cdots\!43}a^{6}+\frac{46\!\cdots\!32}{51\!\cdots\!43}a^{5}+\frac{88\!\cdots\!76}{51\!\cdots\!43}a^{4}+\frac{14\!\cdots\!82}{51\!\cdots\!43}a^{3}+\frac{20\!\cdots\!37}{51\!\cdots\!43}a^{2}+\frac{28\!\cdots\!38}{72\!\cdots\!49}a+\frac{98\!\cdots\!54}{24\!\cdots\!49}$, $\frac{6723749240270}{72\!\cdots\!49}a^{9}-\frac{9969578839580}{16\!\cdots\!43}a^{8}+\frac{707827240917014}{72\!\cdots\!49}a^{7}-\frac{35\!\cdots\!03}{72\!\cdots\!49}a^{6}+\frac{36\!\cdots\!88}{72\!\cdots\!49}a^{5}-\frac{15\!\cdots\!15}{72\!\cdots\!49}a^{4}+\frac{53\!\cdots\!32}{72\!\cdots\!49}a^{3}-\frac{49\!\cdots\!65}{10\!\cdots\!07}a^{2}-\frac{13\!\cdots\!84}{72\!\cdots\!49}a-\frac{11\!\cdots\!48}{24\!\cdots\!49}$ (assuming GRH) | 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: | \( 257.113789169 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 257.113789169 \cdot 42150}{2\cdot\sqrt{1856465000000000000000}}\cr\approx \mathstrut & 1.23154110241 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 10 |
The 10 conjugacy class representatives for $C_{10}$ |
Character table for $C_{10}$ |
Intermediate fields
\(\Q(\sqrt{-26}) \), 5.5.390625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/7.1.0.1}{1} }^{10}$ | ${\href{/padicField/11.10.0.1}{10} }$ | R | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.10.0.1}{10} }$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.1.0.1}{1} }^{10}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.15.13 | $x^{10} - 12 x^{9} + 174 x^{8} - 640 x^{7} - 40 x^{6} + 11424 x^{5} - 7984 x^{4} - 79360 x^{3} + 84112 x^{2} + 955968 x + 1404384$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ |
\(5\) | 5.5.8.2 | $x^{5} + 20 x^{4} + 5$ | $5$ | $1$ | $8$ | $C_5$ | $[2]$ |
5.5.8.2 | $x^{5} + 20 x^{4} + 5$ | $5$ | $1$ | $8$ | $C_5$ | $[2]$ | |
\(13\) | 13.10.5.2 | $x^{10} + 114244 x^{2} - 4084223$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.104.2t1.b.a | $1$ | $ 2^{3} \cdot 13 $ | \(\Q(\sqrt{-26}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.25.5t1.a.b | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.2600.10t1.a.c | $1$ | $ 2^{3} \cdot 5^{2} \cdot 13 $ | 10.0.1856465000000000000000.3 | $C_{10}$ (as 10T1) | $0$ | $-1$ |
* | 1.25.5t1.a.d | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.2600.10t1.a.a | $1$ | $ 2^{3} \cdot 5^{2} \cdot 13 $ | 10.0.1856465000000000000000.3 | $C_{10}$ (as 10T1) | $0$ | $-1$ |
* | 1.25.5t1.a.a | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.2600.10t1.a.d | $1$ | $ 2^{3} \cdot 5^{2} \cdot 13 $ | 10.0.1856465000000000000000.3 | $C_{10}$ (as 10T1) | $0$ | $-1$ |
* | 1.25.5t1.a.c | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.2600.10t1.a.b | $1$ | $ 2^{3} \cdot 5^{2} \cdot 13 $ | 10.0.1856465000000000000000.3 | $C_{10}$ (as 10T1) | $0$ | $-1$ |