Properties

Label 45.45.183...361.1
Degree $45$
Signature $[45, 0]$
Discriminant $1.839\times 10^{91}$
Root discriminant \(106.68\)
Ramified primes $3,31$
Class number not computed
Class group not computed
Galois group $C_3\times C_{15}$ (as 45T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 87*x^43 - 4*x^42 + 3402*x^41 + 300*x^40 - 79106*x^39 - 10008*x^38 + 1219668*x^37 + 196264*x^36 - 13165542*x^35 - 2519472*x^34 + 102436164*x^33 + 22316760*x^32 - 583045167*x^31 - 140145264*x^30 + 2440735434*x^29 + 631757436*x^28 - 7504722910*x^27 - 2050106328*x^26 + 16845457116*x^25 + 4771066968*x^24 - 27354225744*x^23 - 7899876240*x^22 + 31819110888*x^21 + 9218870784*x^20 - 26289431304*x^19 - 7517163552*x^18 + 15324352272*x^17 + 4245866400*x^16 - 6258116631*x^15 - 1643669037*x^14 + 1770836592*x^13 + 429144192*x^12 - 340508538*x^11 - 73586928*x^10 + 43022076*x^9 + 7934841*x^8 - 3375522*x^7 - 502012*x^6 + 149688*x^5 + 16758*x^4 - 3200*x^3 - 252*x^2 + 24*x + 1)
 
gp: K = bnfinit(y^45 - 87*y^43 - 4*y^42 + 3402*y^41 + 300*y^40 - 79106*y^39 - 10008*y^38 + 1219668*y^37 + 196264*y^36 - 13165542*y^35 - 2519472*y^34 + 102436164*y^33 + 22316760*y^32 - 583045167*y^31 - 140145264*y^30 + 2440735434*y^29 + 631757436*y^28 - 7504722910*y^27 - 2050106328*y^26 + 16845457116*y^25 + 4771066968*y^24 - 27354225744*y^23 - 7899876240*y^22 + 31819110888*y^21 + 9218870784*y^20 - 26289431304*y^19 - 7517163552*y^18 + 15324352272*y^17 + 4245866400*y^16 - 6258116631*y^15 - 1643669037*y^14 + 1770836592*y^13 + 429144192*y^12 - 340508538*y^11 - 73586928*y^10 + 43022076*y^9 + 7934841*y^8 - 3375522*y^7 - 502012*y^6 + 149688*y^5 + 16758*y^4 - 3200*y^3 - 252*y^2 + 24*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - 87*x^43 - 4*x^42 + 3402*x^41 + 300*x^40 - 79106*x^39 - 10008*x^38 + 1219668*x^37 + 196264*x^36 - 13165542*x^35 - 2519472*x^34 + 102436164*x^33 + 22316760*x^32 - 583045167*x^31 - 140145264*x^30 + 2440735434*x^29 + 631757436*x^28 - 7504722910*x^27 - 2050106328*x^26 + 16845457116*x^25 + 4771066968*x^24 - 27354225744*x^23 - 7899876240*x^22 + 31819110888*x^21 + 9218870784*x^20 - 26289431304*x^19 - 7517163552*x^18 + 15324352272*x^17 + 4245866400*x^16 - 6258116631*x^15 - 1643669037*x^14 + 1770836592*x^13 + 429144192*x^12 - 340508538*x^11 - 73586928*x^10 + 43022076*x^9 + 7934841*x^8 - 3375522*x^7 - 502012*x^6 + 149688*x^5 + 16758*x^4 - 3200*x^3 - 252*x^2 + 24*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 87*x^43 - 4*x^42 + 3402*x^41 + 300*x^40 - 79106*x^39 - 10008*x^38 + 1219668*x^37 + 196264*x^36 - 13165542*x^35 - 2519472*x^34 + 102436164*x^33 + 22316760*x^32 - 583045167*x^31 - 140145264*x^30 + 2440735434*x^29 + 631757436*x^28 - 7504722910*x^27 - 2050106328*x^26 + 16845457116*x^25 + 4771066968*x^24 - 27354225744*x^23 - 7899876240*x^22 + 31819110888*x^21 + 9218870784*x^20 - 26289431304*x^19 - 7517163552*x^18 + 15324352272*x^17 + 4245866400*x^16 - 6258116631*x^15 - 1643669037*x^14 + 1770836592*x^13 + 429144192*x^12 - 340508538*x^11 - 73586928*x^10 + 43022076*x^9 + 7934841*x^8 - 3375522*x^7 - 502012*x^6 + 149688*x^5 + 16758*x^4 - 3200*x^3 - 252*x^2 + 24*x + 1)
 

\( x^{45} - 87 x^{43} - 4 x^{42} + 3402 x^{41} + 300 x^{40} - 79106 x^{39} - 10008 x^{38} + 1219668 x^{37} + 196264 x^{36} - 13165542 x^{35} - 2519472 x^{34} + 102436164 x^{33} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $45$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[45, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(183\!\cdots\!361\) \(\medspace = 3^{60}\cdot 31^{42}\) 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:  \(106.68\)
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))
 
Ramified primes:   \(3\), \(31\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $45$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(279=3^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{279}(256,·)$, $\chi_{279}(1,·)$, $\chi_{279}(4,·)$, $\chi_{279}(133,·)$, $\chi_{279}(262,·)$, $\chi_{279}(7,·)$, $\chi_{279}(10,·)$, $\chi_{279}(268,·)$, $\chi_{279}(142,·)$, $\chi_{279}(16,·)$, $\chi_{279}(19,·)$, $\chi_{279}(25,·)$, $\chi_{279}(28,·)$, $\chi_{279}(157,·)$, $\chi_{279}(160,·)$, $\chi_{279}(163,·)$, $\chi_{279}(40,·)$, $\chi_{279}(169,·)$, $\chi_{279}(175,·)$, $\chi_{279}(49,·)$, $\chi_{279}(187,·)$, $\chi_{279}(190,·)$, $\chi_{279}(64,·)$, $\chi_{279}(193,·)$, $\chi_{279}(67,·)$, $\chi_{279}(196,·)$, $\chi_{279}(70,·)$, $\chi_{279}(202,·)$, $\chi_{279}(76,·)$, $\chi_{279}(205,·)$, $\chi_{279}(82,·)$, $\chi_{279}(211,·)$, $\chi_{279}(214,·)$, $\chi_{279}(94,·)$, $\chi_{279}(97,·)$, $\chi_{279}(226,·)$, $\chi_{279}(100,·)$, $\chi_{279}(103,·)$, $\chi_{279}(235,·)$, $\chi_{279}(109,·)$, $\chi_{279}(112,·)$, $\chi_{279}(118,·)$, $\chi_{279}(121,·)$, $\chi_{279}(250,·)$, $\chi_{279}(253,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $\frac{1}{5}a^{42}-\frac{1}{5}a^{41}+\frac{1}{5}a^{40}-\frac{1}{5}a^{39}-\frac{1}{5}a^{38}+\frac{1}{5}a^{36}-\frac{2}{5}a^{32}-\frac{2}{5}a^{29}+\frac{2}{5}a^{27}+\frac{2}{5}a^{24}+\frac{1}{5}a^{20}-\frac{1}{5}a^{19}+\frac{1}{5}a^{18}+\frac{2}{5}a^{17}-\frac{2}{5}a^{15}+\frac{1}{5}a^{14}-\frac{1}{5}a^{12}+\frac{1}{5}a^{9}+\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{2785}a^{43}+\frac{35}{557}a^{42}+\frac{161}{557}a^{41}-\frac{189}{557}a^{40}+\frac{453}{2785}a^{39}-\frac{256}{2785}a^{38}-\frac{484}{2785}a^{37}-\frac{889}{2785}a^{36}+\frac{180}{557}a^{35}+\frac{277}{557}a^{34}+\frac{598}{2785}a^{33}+\frac{1363}{2785}a^{32}-\frac{160}{557}a^{31}-\frac{1182}{2785}a^{30}-\frac{892}{2785}a^{29}-\frac{928}{2785}a^{28}+\frac{392}{2785}a^{27}-\frac{23}{557}a^{26}+\frac{1367}{2785}a^{25}+\frac{827}{2785}a^{24}-\frac{106}{557}a^{23}+\frac{175}{557}a^{22}+\frac{606}{2785}a^{21}+\frac{75}{557}a^{20}+\frac{167}{557}a^{19}+\frac{888}{2785}a^{18}-\frac{938}{2785}a^{17}+\frac{8}{2785}a^{16}-\frac{206}{2785}a^{15}-\frac{764}{2785}a^{14}-\frac{296}{2785}a^{13}-\frac{31}{2785}a^{12}+\frac{85}{557}a^{11}-\frac{1289}{2785}a^{10}+\frac{1302}{2785}a^{9}-\frac{726}{2785}a^{8}+\frac{704}{2785}a^{7}-\frac{1012}{2785}a^{6}-\frac{1162}{2785}a^{5}+\frac{1013}{2785}a^{4}-\frac{403}{2785}a^{3}+\frac{508}{2785}a^{2}+\frac{137}{2785}a+\frac{134}{2785}$, $\frac{1}{66\!\cdots\!65}a^{44}-\frac{52\!\cdots\!62}{66\!\cdots\!65}a^{43}-\frac{55\!\cdots\!32}{66\!\cdots\!65}a^{42}-\frac{10\!\cdots\!78}{66\!\cdots\!65}a^{41}+\frac{61\!\cdots\!06}{66\!\cdots\!65}a^{40}+\frac{46\!\cdots\!61}{13\!\cdots\!73}a^{39}-\frac{33\!\cdots\!12}{13\!\cdots\!73}a^{38}+\frac{29\!\cdots\!74}{66\!\cdots\!65}a^{37}+\frac{66\!\cdots\!16}{66\!\cdots\!65}a^{36}-\frac{50\!\cdots\!90}{13\!\cdots\!73}a^{35}+\frac{27\!\cdots\!88}{66\!\cdots\!65}a^{34}+\frac{80\!\cdots\!22}{66\!\cdots\!65}a^{33}+\frac{30\!\cdots\!58}{66\!\cdots\!65}a^{32}+\frac{29\!\cdots\!98}{66\!\cdots\!65}a^{31}+\frac{87\!\cdots\!32}{66\!\cdots\!65}a^{30}+\frac{36\!\cdots\!52}{13\!\cdots\!73}a^{29}-\frac{28\!\cdots\!02}{66\!\cdots\!65}a^{28}-\frac{34\!\cdots\!53}{66\!\cdots\!65}a^{27}-\frac{16\!\cdots\!17}{43\!\cdots\!85}a^{26}-\frac{32\!\cdots\!77}{66\!\cdots\!65}a^{25}+\frac{10\!\cdots\!72}{66\!\cdots\!65}a^{24}+\frac{34\!\cdots\!85}{13\!\cdots\!73}a^{23}+\frac{26\!\cdots\!66}{66\!\cdots\!65}a^{22}+\frac{26\!\cdots\!93}{66\!\cdots\!65}a^{21}+\frac{32\!\cdots\!03}{66\!\cdots\!65}a^{20}-\frac{20\!\cdots\!45}{13\!\cdots\!73}a^{19}+\frac{30\!\cdots\!04}{66\!\cdots\!65}a^{18}+\frac{29\!\cdots\!07}{13\!\cdots\!73}a^{17}-\frac{10\!\cdots\!82}{66\!\cdots\!65}a^{16}+\frac{25\!\cdots\!17}{66\!\cdots\!65}a^{15}-\frac{42\!\cdots\!19}{13\!\cdots\!73}a^{14}-\frac{60\!\cdots\!04}{66\!\cdots\!65}a^{13}+\frac{13\!\cdots\!79}{66\!\cdots\!65}a^{12}-\frac{12\!\cdots\!54}{66\!\cdots\!65}a^{11}-\frac{44\!\cdots\!75}{13\!\cdots\!73}a^{10}+\frac{12\!\cdots\!88}{66\!\cdots\!65}a^{9}-\frac{69\!\cdots\!91}{66\!\cdots\!65}a^{8}+\frac{23\!\cdots\!44}{66\!\cdots\!65}a^{7}-\frac{21\!\cdots\!01}{13\!\cdots\!73}a^{6}+\frac{20\!\cdots\!03}{66\!\cdots\!65}a^{5}+\frac{12\!\cdots\!54}{66\!\cdots\!65}a^{4}-\frac{45\!\cdots\!24}{13\!\cdots\!73}a^{3}+\frac{10\!\cdots\!21}{66\!\cdots\!65}a^{2}+\frac{14\!\cdots\!99}{66\!\cdots\!65}a+\frac{11\!\cdots\!74}{66\!\cdots\!65}$ 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

not computed

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:  $44$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^45 - 87*x^43 - 4*x^42 + 3402*x^41 + 300*x^40 - 79106*x^39 - 10008*x^38 + 1219668*x^37 + 196264*x^36 - 13165542*x^35 - 2519472*x^34 + 102436164*x^33 + 22316760*x^32 - 583045167*x^31 - 140145264*x^30 + 2440735434*x^29 + 631757436*x^28 - 7504722910*x^27 - 2050106328*x^26 + 16845457116*x^25 + 4771066968*x^24 - 27354225744*x^23 - 7899876240*x^22 + 31819110888*x^21 + 9218870784*x^20 - 26289431304*x^19 - 7517163552*x^18 + 15324352272*x^17 + 4245866400*x^16 - 6258116631*x^15 - 1643669037*x^14 + 1770836592*x^13 + 429144192*x^12 - 340508538*x^11 - 73586928*x^10 + 43022076*x^9 + 7934841*x^8 - 3375522*x^7 - 502012*x^6 + 149688*x^5 + 16758*x^4 - 3200*x^3 - 252*x^2 + 24*x + 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^45 - 87*x^43 - 4*x^42 + 3402*x^41 + 300*x^40 - 79106*x^39 - 10008*x^38 + 1219668*x^37 + 196264*x^36 - 13165542*x^35 - 2519472*x^34 + 102436164*x^33 + 22316760*x^32 - 583045167*x^31 - 140145264*x^30 + 2440735434*x^29 + 631757436*x^28 - 7504722910*x^27 - 2050106328*x^26 + 16845457116*x^25 + 4771066968*x^24 - 27354225744*x^23 - 7899876240*x^22 + 31819110888*x^21 + 9218870784*x^20 - 26289431304*x^19 - 7517163552*x^18 + 15324352272*x^17 + 4245866400*x^16 - 6258116631*x^15 - 1643669037*x^14 + 1770836592*x^13 + 429144192*x^12 - 340508538*x^11 - 73586928*x^10 + 43022076*x^9 + 7934841*x^8 - 3375522*x^7 - 502012*x^6 + 149688*x^5 + 16758*x^4 - 3200*x^3 - 252*x^2 + 24*x + 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^45 - 87*x^43 - 4*x^42 + 3402*x^41 + 300*x^40 - 79106*x^39 - 10008*x^38 + 1219668*x^37 + 196264*x^36 - 13165542*x^35 - 2519472*x^34 + 102436164*x^33 + 22316760*x^32 - 583045167*x^31 - 140145264*x^30 + 2440735434*x^29 + 631757436*x^28 - 7504722910*x^27 - 2050106328*x^26 + 16845457116*x^25 + 4771066968*x^24 - 27354225744*x^23 - 7899876240*x^22 + 31819110888*x^21 + 9218870784*x^20 - 26289431304*x^19 - 7517163552*x^18 + 15324352272*x^17 + 4245866400*x^16 - 6258116631*x^15 - 1643669037*x^14 + 1770836592*x^13 + 429144192*x^12 - 340508538*x^11 - 73586928*x^10 + 43022076*x^9 + 7934841*x^8 - 3375522*x^7 - 502012*x^6 + 149688*x^5 + 16758*x^4 - 3200*x^3 - 252*x^2 + 24*x + 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^45 - 87*x^43 - 4*x^42 + 3402*x^41 + 300*x^40 - 79106*x^39 - 10008*x^38 + 1219668*x^37 + 196264*x^36 - 13165542*x^35 - 2519472*x^34 + 102436164*x^33 + 22316760*x^32 - 583045167*x^31 - 140145264*x^30 + 2440735434*x^29 + 631757436*x^28 - 7504722910*x^27 - 2050106328*x^26 + 16845457116*x^25 + 4771066968*x^24 - 27354225744*x^23 - 7899876240*x^22 + 31819110888*x^21 + 9218870784*x^20 - 26289431304*x^19 - 7517163552*x^18 + 15324352272*x^17 + 4245866400*x^16 - 6258116631*x^15 - 1643669037*x^14 + 1770836592*x^13 + 429144192*x^12 - 340508538*x^11 - 73586928*x^10 + 43022076*x^9 + 7934841*x^8 - 3375522*x^7 - 502012*x^6 + 149688*x^5 + 16758*x^4 - 3200*x^3 - 252*x^2 + 24*x + 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

$C_3\times C_{15}$ (as 45T2):

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 45
The 45 conjugacy class representatives for $C_3\times C_{15}$
Character table for $C_3\times C_{15}$ is not computed

Intermediate fields

3.3.961.1, 3.3.77841.2, 3.3.77841.1, \(\Q(\zeta_{9})^+\), 5.5.923521.1, 9.9.471655843734321.1, \(\Q(\zeta_{31})^+\), 15.15.2639300305759427100493462112721.1, 15.15.2639300305759427100493462112721.2, 15.15.2746410307762150989067078161.1

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 $15^{3}$ R ${\href{/padicField/5.3.0.1}{3} }^{15}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ R ${\href{/padicField/37.3.0.1}{3} }^{15}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$

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
\(3\) Copy content Toggle raw display Deg $45$$3$$15$$60$
\(31\) Copy content Toggle raw display Deg $45$$15$$3$$42$