Normalized defining polynomial
\( x^{25} + 4x - 4 \)
Invariants
Degree: | $25$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1579621612245345690669012483485992656830464\) \(\medspace = 2^{24}\cdot 8054869\cdot 11688928159776857914348061941\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(48.75\) | 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: | $2^{24/25}8054869^{1/2}11688928159776857914348061941^{1/2}\approx 5.969052262562249e+17$ | ||
Ramified primes: | \(2\), \(8054869\), \(11688928159776857914348061941\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{94152\!\cdots\!40729}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{2}a^{16}$, $\frac{1}{2}a^{17}$, $\frac{1}{2}a^{18}$, $\frac{1}{2}a^{19}$, $\frac{1}{2}a^{20}$, $\frac{1}{2}a^{21}$, $\frac{1}{2}a^{22}$, $\frac{1}{2}a^{23}$, $\frac{1}{2}a^{24}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $12$ | 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: | $a-1$, $\frac{1}{2}a^{18}+\frac{1}{2}a^{17}+\frac{1}{2}a^{16}+\frac{1}{2}a^{15}+\frac{1}{2}a^{14}+\frac{1}{2}a^{13}+1$, $\frac{1}{2}a^{20}+\frac{1}{2}a^{15}-a+1$, $\frac{1}{2}a^{20}+\frac{1}{2}a^{18}+\frac{1}{2}a^{16}+\frac{1}{2}a^{15}+\frac{1}{2}a^{14}+\frac{1}{2}a^{13}+a^{11}+a^{9}+a^{4}-a^{3}+a^{2}-a+1$, $\frac{1}{2}a^{20}+\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{13}+a^{7}+a^{6}-a^{3}-a^{2}+1$, $a^{24}+a^{23}+a^{22}+\frac{1}{2}a^{21}+\frac{1}{2}a^{20}+\frac{1}{2}a^{19}+\frac{1}{2}a^{15}+a^{14}+a^{13}+a^{12}-a^{8}+a^{5}+2a^{4}+a^{3}-2a+3$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{22}+\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-a^{19}-a^{17}-a^{16}-\frac{1}{2}a^{15}-2a^{14}-\frac{1}{2}a^{13}-2a^{11}-a^{8}+a^{7}-a^{6}-a^{5}+3a^{4}-a^{3}+3a+1$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{23}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{3}{2}a^{20}-\frac{3}{2}a^{18}+\frac{1}{2}a^{17}-\frac{1}{2}a^{16}+\frac{1}{2}a^{15}+a^{14}+2a^{12}-a^{11}+2a^{10}-2a^{9}+a^{8}-2a^{7}-a^{6}-3a^{4}+3a^{3}-4a^{2}+5a-1$, $\frac{1}{2}a^{24}-a^{23}+\frac{1}{2}a^{22}+\frac{3}{2}a^{21}-\frac{1}{2}a^{20}-a^{19}+a^{18}+a^{17}-\frac{3}{2}a^{16}-a^{15}+2a^{14}-3a^{12}+a^{11}+3a^{10}-2a^{9}-2a^{8}+3a^{7}+2a^{6}-3a^{5}-a^{4}+5a^{3}-6a+5$, $a^{24}+\frac{1}{2}a^{21}+a^{18}+\frac{1}{2}a^{17}+a^{15}-\frac{1}{2}a^{13}+a^{12}-a^{10}+a^{9}+a^{8}-a^{7}+a^{6}+2a^{5}-2a^{4}-a^{3}+2a^{2}-2a+3$, $\frac{1}{2}a^{24}+\frac{1}{2}a^{23}+\frac{1}{2}a^{22}+a^{20}+\frac{3}{2}a^{19}+a^{18}+a^{17}+\frac{3}{2}a^{16}+\frac{1}{2}a^{13}+2a^{10}+a^{9}+a^{7}-2a^{5}+a^{3}-a^{2}+2a+5$, $\frac{1}{2}a^{23}+a^{22}+\frac{3}{2}a^{21}+a^{20}+\frac{3}{2}a^{19}+2a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+\frac{1}{2}a^{13}+a^{11}+a^{8}-a^{5}-3a^{2}-a-1$ (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: | \( 248830673715.64108 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 248830673715.64108 \cdot 1}{2\cdot\sqrt{1579621612245345690669012483485992656830464}}\cr\approx \mathstrut & 0.749524446172306 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 15511210043330985984000000 |
The 1958 conjugacy class representatives for $S_{25}$ |
Character table for $S_{25}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | $19{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }$ | $16{,}\,{\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | $22{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $25$ | $25$ | $1$ | $24$ | |||
\(8054869\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(116\!\cdots\!941\) | $\Q_{11\!\cdots\!41}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{11\!\cdots\!41}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ |