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Properties

Label 40.0.324...000.1
Degree $40$
Signature $[0, 20]$
Discriminant $3.241\times 10^{63}$
Root discriminant \(38.71\)
Ramified primes $2,5,11$
Class number $62$ (GRH)
Class group [62] (GRH)
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 3*x^38 + 8*x^36 - 21*x^34 + 55*x^32 - 144*x^30 + 377*x^28 - 987*x^26 + 2584*x^24 - 6765*x^22 + 17711*x^20 - 6765*x^18 + 2584*x^16 - 987*x^14 + 377*x^12 - 144*x^10 + 55*x^8 - 21*x^6 + 8*x^4 - 3*x^2 + 1)
 
gp: K = bnfinit(y^40 - 3*y^38 + 8*y^36 - 21*y^34 + 55*y^32 - 144*y^30 + 377*y^28 - 987*y^26 + 2584*y^24 - 6765*y^22 + 17711*y^20 - 6765*y^18 + 2584*y^16 - 987*y^14 + 377*y^12 - 144*y^10 + 55*y^8 - 21*y^6 + 8*y^4 - 3*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - 3*x^38 + 8*x^36 - 21*x^34 + 55*x^32 - 144*x^30 + 377*x^28 - 987*x^26 + 2584*x^24 - 6765*x^22 + 17711*x^20 - 6765*x^18 + 2584*x^16 - 987*x^14 + 377*x^12 - 144*x^10 + 55*x^8 - 21*x^6 + 8*x^4 - 3*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - 3*x^38 + 8*x^36 - 21*x^34 + 55*x^32 - 144*x^30 + 377*x^28 - 987*x^26 + 2584*x^24 - 6765*x^22 + 17711*x^20 - 6765*x^18 + 2584*x^16 - 987*x^14 + 377*x^12 - 144*x^10 + 55*x^8 - 21*x^6 + 8*x^4 - 3*x^2 + 1)
 

\( x^{40} - 3 x^{38} + 8 x^{36} - 21 x^{34} + 55 x^{32} - 144 x^{30} + 377 x^{28} - 987 x^{26} + 2584 x^{24} + \cdots + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $40$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 20]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(3241429490243539842962382172914969829546393600000000000000000000\) \(\medspace = 2^{40}\cdot 5^{20}\cdot 11^{36}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(38.71\)
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:  $2\cdot 5^{1/2}11^{9/10}\approx 38.70511966206675$
Ramified primes:   \(2\), \(5\), \(11\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $40$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(220=2^{2}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(131,·)$, $\chi_{220}(179,·)$, $\chi_{220}(129,·)$, $\chi_{220}(9,·)$, $\chi_{220}(139,·)$, $\chi_{220}(141,·)$, $\chi_{220}(19,·)$, $\chi_{220}(21,·)$, $\chi_{220}(151,·)$, $\chi_{220}(29,·)$, $\chi_{220}(159,·)$, $\chi_{220}(161,·)$, $\chi_{220}(91,·)$, $\chi_{220}(39,·)$, $\chi_{220}(41,·)$, $\chi_{220}(171,·)$, $\chi_{220}(49,·)$, $\chi_{220}(51,·)$, $\chi_{220}(181,·)$, $\chi_{220}(31,·)$, $\chi_{220}(61,·)$, $\chi_{220}(191,·)$, $\chi_{220}(69,·)$, $\chi_{220}(71,·)$, $\chi_{220}(201,·)$, $\chi_{220}(119,·)$, $\chi_{220}(79,·)$, $\chi_{220}(81,·)$, $\chi_{220}(211,·)$, $\chi_{220}(89,·)$, $\chi_{220}(219,·)$, $\chi_{220}(199,·)$, $\chi_{220}(59,·)$, $\chi_{220}(101,·)$, $\chi_{220}(109,·)$, $\chi_{220}(111,·)$, $\chi_{220}(189,·)$, $\chi_{220}(169,·)$, $\chi_{220}(149,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{524288}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{17711}a^{22}-\frac{6765}{17711}$, $\frac{1}{17711}a^{23}-\frac{6765}{17711}a$, $\frac{1}{17711}a^{24}-\frac{6765}{17711}a^{2}$, $\frac{1}{17711}a^{25}-\frac{6765}{17711}a^{3}$, $\frac{1}{17711}a^{26}-\frac{6765}{17711}a^{4}$, $\frac{1}{17711}a^{27}-\frac{6765}{17711}a^{5}$, $\frac{1}{17711}a^{28}-\frac{6765}{17711}a^{6}$, $\frac{1}{17711}a^{29}-\frac{6765}{17711}a^{7}$, $\frac{1}{17711}a^{30}-\frac{6765}{17711}a^{8}$, $\frac{1}{17711}a^{31}-\frac{6765}{17711}a^{9}$, $\frac{1}{17711}a^{32}-\frac{6765}{17711}a^{10}$, $\frac{1}{17711}a^{33}-\frac{6765}{17711}a^{11}$, $\frac{1}{17711}a^{34}-\frac{6765}{17711}a^{12}$, $\frac{1}{17711}a^{35}-\frac{6765}{17711}a^{13}$, $\frac{1}{17711}a^{36}-\frac{6765}{17711}a^{14}$, $\frac{1}{17711}a^{37}-\frac{6765}{17711}a^{15}$, $\frac{1}{17711}a^{38}-\frac{6765}{17711}a^{16}$, $\frac{1}{17711}a^{39}-\frac{6765}{17711}a^{17}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{62}$, which has order $62$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $19$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{1597}{17711} a^{39} + \frac{63245986}{17711} a^{17} \)  (order $44$) Copy content Toggle raw display
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:   $\frac{1}{17711}a^{23}+\frac{46368}{17711}a$, $\frac{6765}{17711}a^{38}-\frac{20672}{17711}a^{36}+\frac{54120}{17711}a^{34}-\frac{142065}{17711}a^{32}+\frac{372075}{17711}a^{30}-\frac{974160}{17711}a^{28}+\frac{2550405}{17711}a^{26}-\frac{6677055}{17711}a^{24}+\frac{17480760}{17711}a^{22}-2584a^{20}+6765a^{18}-\frac{45765225}{17711}a^{16}+\frac{2550408}{17711}a^{14}-\frac{6677055}{17711}a^{12}+\frac{2550405}{17711}a^{10}-\frac{974160}{17711}a^{8}+\frac{372075}{17711}a^{6}-\frac{142065}{17711}a^{4}+\frac{54120}{17711}a^{2}-\frac{20295}{17711}$, $\frac{6765}{17711}a^{38}-\frac{20295}{17711}a^{36}+\frac{54120}{17711}a^{34}-\frac{142065}{17711}a^{32}+\frac{372075}{17711}a^{30}-\frac{974168}{17711}a^{28}+\frac{2550405}{17711}a^{26}-\frac{6677055}{17711}a^{24}+\frac{17480760}{17711}a^{22}-2584a^{20}+6765a^{18}-\frac{45765225}{17711}a^{16}+\frac{17480760}{17711}a^{14}-\frac{6677055}{17711}a^{12}+\frac{2550405}{17711}a^{10}-\frac{974160}{17711}a^{8}+\frac{54264}{17711}a^{6}-\frac{142065}{17711}a^{4}+\frac{54120}{17711}a^{2}-\frac{20295}{17711}$, $\frac{55}{17711}a^{32}+\frac{2178309}{17711}a^{10}-1$, $\frac{144}{17711}a^{34}-\frac{8}{17711}a^{28}+\frac{5702887}{17711}a^{12}-\frac{317811}{17711}a^{6}+1$, $\frac{13}{17711}a^{29}+\frac{514229}{17711}a^{7}+1$, $\frac{34}{17711}a^{31}+\frac{21}{17711}a^{30}+\frac{1346269}{17711}a^{9}+\frac{832040}{17711}a^{8}$, $\frac{1597}{17711}a^{39}-\frac{6765}{17711}a^{38}+\frac{20295}{17711}a^{36}-\frac{54120}{17711}a^{34}+\frac{142065}{17711}a^{32}-\frac{372075}{17711}a^{30}+\frac{974160}{17711}a^{28}-\frac{2550405}{17711}a^{26}+\frac{6677055}{17711}a^{24}-\frac{17480760}{17711}a^{22}+2584a^{20}-6765a^{18}+\frac{63245986}{17711}a^{17}+\frac{45765225}{17711}a^{16}-\frac{17480760}{17711}a^{14}+\frac{6677055}{17711}a^{12}-\frac{2550405}{17711}a^{10}+\frac{974160}{17711}a^{8}-\frac{372075}{17711}a^{6}+\frac{142065}{17711}a^{4}-\frac{54120}{17711}a^{2}+\frac{20295}{17711}$, $\frac{34}{17711}a^{31}+\frac{1346269}{17711}a^{9}-1$, $\frac{10946}{17711}a^{39}-\frac{32838}{17711}a^{37}+\frac{87568}{17711}a^{35}-\frac{229866}{17711}a^{33}+\frac{602030}{17711}a^{31}-\frac{1576224}{17711}a^{29}+\frac{4126642}{17711}a^{27}+\frac{3}{17711}a^{26}-\frac{10803702}{17711}a^{25}+\frac{28284464}{17711}a^{23}-4181a^{21}+10946a^{19}-\frac{74049690}{17711}a^{17}+\frac{28284464}{17711}a^{15}-\frac{10803702}{17711}a^{13}+\frac{4126642}{17711}a^{11}-\frac{1576224}{17711}a^{9}+\frac{602030}{17711}a^{7}-\frac{229866}{17711}a^{5}+\frac{121393}{17711}a^{4}+\frac{87568}{17711}a^{3}-\frac{32838}{17711}a$, $\frac{6765}{17711}a^{38}-\frac{20295}{17711}a^{36}+\frac{54120}{17711}a^{34}-\frac{142065}{17711}a^{32}+\frac{372075}{17711}a^{30}-\frac{974168}{17711}a^{28}+\frac{2550405}{17711}a^{26}-\frac{6677057}{17711}a^{24}+\frac{17480760}{17711}a^{22}-2584a^{20}+6765a^{18}-\frac{45765225}{17711}a^{16}+\frac{17480760}{17711}a^{14}-\frac{6677055}{17711}a^{12}+\frac{2550405}{17711}a^{10}-\frac{974160}{17711}a^{8}+\frac{54264}{17711}a^{6}-\frac{142065}{17711}a^{4}-\frac{20905}{17711}a^{2}-\frac{20295}{17711}$, $\frac{4791}{17711}a^{38}-\frac{13153}{17711}a^{36}+\frac{33681}{17711}a^{34}-\frac{87890}{17711}a^{32}+\frac{229989}{17711}a^{30}-\frac{602077}{17711}a^{28}+\frac{1576242}{17711}a^{26}-\frac{4126649}{17711}a^{24}+\frac{10803705}{17711}a^{22}-1597a^{20}+4181a^{18}-\frac{4126648}{17711}a^{16}-\frac{13354113}{17711}a^{14}+\frac{5100818}{17711}a^{12}-\frac{1948341}{17711}a^{10}+\frac{744205}{17711}a^{8}-\frac{284274}{17711}a^{6}+\frac{108617}{17711}a^{4}-\frac{41577}{17711}a^{2}+\frac{16114}{17711}$, $\frac{6765}{17711}a^{38}-\frac{20295}{17711}a^{36}+\frac{54353}{17711}a^{34}-\frac{142065}{17711}a^{32}+\frac{372075}{17711}a^{30}-\frac{974160}{17711}a^{28}+\frac{2550408}{17711}a^{26}-\frac{6677055}{17711}a^{24}+\frac{17480760}{17711}a^{22}-2584a^{20}+6765a^{18}-\frac{45765225}{17711}a^{16}+\frac{17480760}{17711}a^{14}+\frac{2550410}{17711}a^{12}+\frac{2550405}{17711}a^{10}-\frac{974160}{17711}a^{8}+\frac{372075}{17711}a^{6}-\frac{20672}{17711}a^{4}+\frac{54120}{17711}a^{2}-\frac{20295}{17711}$, $\frac{7752}{17711}a^{39}-\frac{20672}{17711}a^{37}+\frac{54264}{17711}a^{35}-\frac{142120}{17711}a^{33}+\frac{1870}{89}a^{31}-\frac{974181}{17711}a^{29}+\frac{2550408}{17711}a^{27}-\frac{6677056}{17711}a^{25}+\frac{17480760}{17711}a^{23}-2584a^{21}+6765a^{19}-\frac{6677056}{17711}a^{17}+\frac{2550408}{17711}a^{15}-\frac{974168}{17711}a^{13}+\frac{372096}{17711}a^{11}+\frac{6051}{89}a^{9}-\frac{459965}{17711}a^{7}-\frac{20672}{17711}a^{5}+\frac{7752}{17711}a^{3}-\frac{2584}{17711}a$, $\frac{610}{17711}a^{37}-\frac{13}{17711}a^{29}-\frac{1}{17711}a^{22}+\frac{24157817}{17711}a^{15}-\frac{514229}{17711}a^{7}-\frac{10946}{17711}$, $\frac{8}{17711}a^{28}-\frac{3}{17711}a^{27}+\frac{3}{17711}a^{26}+\frac{317811}{17711}a^{6}-\frac{121393}{17711}a^{5}+\frac{121393}{17711}a^{4}$, $\frac{12543}{17711}a^{38}-\frac{33448}{17711}a^{36}+\frac{233}{17711}a^{35}+\frac{87801}{17711}a^{34}-\frac{229955}{17711}a^{32}+\frac{602064}{17711}a^{30}-\frac{1576237}{17711}a^{28}+\frac{4126647}{17711}a^{26}-\frac{10803704}{17711}a^{24}-\frac{1}{17711}a^{23}+\frac{28284465}{17711}a^{22}-4181a^{20}+10946a^{18}-\frac{10803704}{17711}a^{16}+\frac{4126647}{17711}a^{14}+\frac{9227465}{17711}a^{13}-\frac{1576237}{17711}a^{12}+\frac{602064}{17711}a^{10}-\frac{229955}{17711}a^{8}+\frac{87801}{17711}a^{6}-\frac{33448}{17711}a^{4}+\frac{12543}{17711}a^{2}-\frac{28657}{17711}a-\frac{4181}{17711}$, $\frac{377}{17711}a^{36}-\frac{55}{17711}a^{33}+\frac{21}{17711}a^{30}+\frac{14930352}{17711}a^{14}-\frac{2178309}{17711}a^{11}+\frac{832040}{17711}a^{8}$, $\frac{987}{17711}a^{37}+\frac{8}{17711}a^{28}-\frac{1}{17711}a^{24}+\frac{39088169}{17711}a^{15}+\frac{317811}{17711}a^{6}-\frac{46368}{17711}a^{2}$ Copy content Toggle raw display (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:  \( 645826241875575.0 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 645826241875575.0 \cdot 62}{44\cdot\sqrt{3241429490243539842962382172914969829546393600000000000000000000}}\cr\approx \mathstrut & 0.146988642294531 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^40 - 3*x^38 + 8*x^36 - 21*x^34 + 55*x^32 - 144*x^30 + 377*x^28 - 987*x^26 + 2584*x^24 - 6765*x^22 + 17711*x^20 - 6765*x^18 + 2584*x^16 - 987*x^14 + 377*x^12 - 144*x^10 + 55*x^8 - 21*x^6 + 8*x^4 - 3*x^2 + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^40 - 3*x^38 + 8*x^36 - 21*x^34 + 55*x^32 - 144*x^30 + 377*x^28 - 987*x^26 + 2584*x^24 - 6765*x^22 + 17711*x^20 - 6765*x^18 + 2584*x^16 - 987*x^14 + 377*x^12 - 144*x^10 + 55*x^8 - 21*x^6 + 8*x^4 - 3*x^2 + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^40 - 3*x^38 + 8*x^36 - 21*x^34 + 55*x^32 - 144*x^30 + 377*x^28 - 987*x^26 + 2584*x^24 - 6765*x^22 + 17711*x^20 - 6765*x^18 + 2584*x^16 - 987*x^14 + 377*x^12 - 144*x^10 + 55*x^8 - 21*x^6 + 8*x^4 - 3*x^2 + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - 3*x^38 + 8*x^36 - 21*x^34 + 55*x^32 - 144*x^30 + 377*x^28 - 987*x^26 + 2584*x^24 - 6765*x^22 + 17711*x^20 - 6765*x^18 + 2584*x^16 - 987*x^14 + 377*x^12 - 144*x^10 + 55*x^8 - 21*x^6 + 8*x^4 - 3*x^2 + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2^2\times C_{10}$ (as 40T7):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{11}) \), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{55})\), \(\Q(i, \sqrt{11})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{5}, \sqrt{11})\), \(\Q(\sqrt{-5}, \sqrt{11})\), \(\Q(\sqrt{-5}, \sqrt{-11})\), \(\Q(\zeta_{11})^+\), 8.0.2342560000.1, 10.0.219503494144.1, 10.10.669871503125.1, 10.0.685948419200000.1, 10.0.7368586534375.1, 10.10.7545432611200000.1, \(\Q(\zeta_{11})\), \(\Q(\zeta_{44})^+\), 20.0.470525233802978928640000000000.1, 20.0.56933553290160450365440000000000.2, \(\Q(\zeta_{44})\), 20.0.54296067514572573056640625.1, 20.20.56933553290160450365440000000000.1, 20.0.56933553290160450365440000000000.1, 20.0.56933553290160450365440000000000.3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }^{4}$ R ${\href{/padicField/7.10.0.1}{10} }^{4}$ R ${\href{/padicField/13.10.0.1}{10} }^{4}$ ${\href{/padicField/17.10.0.1}{10} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{20}$ ${\href{/padicField/29.10.0.1}{10} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{4}$ ${\href{/padicField/37.10.0.1}{10} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{20}$ ${\href{/padicField/47.10.0.1}{10} }^{4}$ ${\href{/padicField/53.10.0.1}{10} }^{4}$ ${\href{/padicField/59.10.0.1}{10} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $20$$2$$10$$20$
Deg $20$$2$$10$$20$
\(5\) Copy content Toggle raw display 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}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
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}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(11\) Copy content Toggle raw display 11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$
11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$