Properties

Label 338.d2
Conductor 338338
Discriminant 17576-17576
j-invariant 13318 \frac{1331}{8}
CM no
Rank 11
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+xy+y=x3+x2+3x5y^2+xy+y=x^3+x^2+3x-5 Copy content Toggle raw display (homogenize, simplify)
y2z+xyz+yz2=x3+x2z+3xz25z3y^2z+xyz+yz^2=x^3+x^2z+3xz^2-5z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x3+3861x282906y^2=x^3+3861x-282906 Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([1, 1, 1, 3, -5])
 
Copy content gp:E = ellinit([1, 1, 1, 3, -5])
 
Copy content magma:E := EllipticCurve([1, 1, 1, 3, -5]);
 
Copy content oscar:E = elliptic_curve([1, 1, 1, 3, -5])
 
Copy content comment:Simplified equation
 
Copy content sage:E.short_weierstrass_model()
 
Copy content magma:WeierstrassModel(E);
 
Copy content oscar:short_weierstrass_model(E)
 

Mordell-Weil group structure

Z\Z

Copy content comment:Mordell-Weil group
 
Copy content magma:MordellWeilGroup(E);
 

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(5,10)(5, 10)0.153682750909108438176744878120.15368275090910843817674487812\infty

Integral points

(1,0) \left(1, 0\right) , (1,2) \left(1, -2\right) , (5,10) \left(5, 10\right) , (5,16) \left(5, -16\right) , (17,64) \left(17, 64\right) , (17,82) \left(17, -82\right) Copy content Toggle raw display

Copy content comment:Integral points
 
Copy content sage:E.integral_points()
 
Copy content magma:IntegralPoints(E);
 

Invariants

Conductor: NN  =  338 338  = 21322 \cdot 13^{2}
Copy content comment:Conductor
 
Copy content sage:E.conductor().factor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Copy content oscar:conductor(E)
 
Discriminant: Δ\Delta  =  17576-17576 = 123133-1 \cdot 2^{3} \cdot 13^{3}
Copy content comment:Discriminant
 
Copy content sage:E.discriminant().factor()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Copy content oscar:discriminant(E)
 
j-invariant: jj  =  13318 \frac{1331}{8}  = 231132^{-3} \cdot 11^{3}
Copy content comment:j-invariant
 
Copy content sage:E.j_invariant().factor()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Copy content oscar:j_invariant(E)
 
Endomorphism ring: End(E)\mathrm{End}(E) = Z\Z
Geometric endomorphism ring: End(EQ)\mathrm{End}(E_{\overline{\Q}})  =  Z\Z    (no potential complex multiplication)
Copy content comment:Potential complex multiplication
 
Copy content sage:E.has_cm()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = SU(2)\mathrm{SU}(2)
Faltings height: hFaltingsh_{\mathrm{Faltings}} ≈ 0.50357809758968124519938431369-0.50357809758968124519938431369
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 1.1448154369550654292127561741-1.1448154369550654292127561741
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
abcabc quality: QQ ≈ 0.93576837136709250.9357683713670925
Szpiro ratio: σm\sigma_{m} ≈ 2.941376754750052.94137675475005

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 1 1
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: rr = 1 1
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) ≈ 0.153682750909108438176744878120.15368275090910843817674487812
Copy content comment:Regulator
 
Copy content sage:E.regulator()
 
Copy content gp:G = E.gen \\ if available matdet(ellheightmatrix(E,G))
 
Copy content magma:Regulator(E);
 
Real period: Ω\Omega ≈ 1.94565404089013374557598260911.9456540408901337455759826091
Copy content comment:Real Period
 
Copy content sage:E.period_lattice().omega()
 
Copy content gp:if(E.disc>0,2,1)*E.omega[1]
 
Copy content magma:(Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: pcp\prod_{p}c_p = 6 6  = 32 3\cdot2
Copy content comment:Tamagawa numbers
 
Copy content sage:E.tamagawa_numbers()
 
Copy content gp:gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Copy content magma:TamagawaNumbers(E);
 
Copy content oscar:tamagawa_numbers(E)
 
Torsion order: #E(Q)tor\#E(\Q)_{\mathrm{tor}} = 11
Copy content comment:Torsion order
 
Copy content sage:E.torsion_order()
 
Copy content gp:elltors(E)[1]
 
Copy content magma:Order(TorsionSubgroup(E));
 
Copy content oscar:prod(torsion_structure(E)[1])
 
Special value: L(E,1) L'(E,1) ≈ 1.79408079192851224940291608011.7940807919285122494029160801
Copy content comment:Special L-value
 
Copy content sage:r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
Copy content gp:[r,L1r] = ellanalyticrank(E); L1r/r!
 
Copy content magma:Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Шan{}_{\mathrm{an}}  ≈  11    (rounded)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

1.794080792L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor211.9456540.1536836121.794080792\begin{aligned} 1.794080792 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.945654 \cdot 0.153683 \cdot 6}{1^2} \\ & \approx 1.794080792\end{aligned}

Copy content comment:BSD formula
 
Copy content sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha) E = EllipticCurve([1, 1, 1, 3, -5]); r = E.rank(); ar = E.analytic_rank(); assert r == ar; Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical(); omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order(); assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
 
Copy content magma:/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */ E := EllipticCurve([1, 1, 1, 3, -5]); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar; sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1); reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E); assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
 

Modular invariants

Modular form   338.2.a.d

q+q2q3+q43q5q63q7+q82q93q10q123q14+3q15+q163q172q186q19+O(q20) q + q^{2} - q^{3} + q^{4} - 3 q^{5} - q^{6} - 3 q^{7} + q^{8} - 2 q^{9} - 3 q^{10} - q^{12} - 3 q^{14} + 3 q^{15} + q^{16} - 3 q^{17} - 2 q^{18} - 6 q^{19} + O(q^{20}) Copy content Toggle raw display

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:\\ actual modular form, use for small N [mf,F] = mffromell(E) Ser(mfcoefs(mf,20),q) \\ or just the series Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 

For more coefficients, see the Downloads section to the right.

Modular degree: 24
Copy content comment:Modular degree
 
Copy content sage:E.modular_degree()
 
Copy content gp:ellmoddegree(E)
 
Copy content magma:ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: yes
Manin constant: 1
Copy content comment:Manin constant
 
Copy content magma:ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 2 primes pp of bad reduction:

pp Tamagawa number Kodaira symbol Reduction type Root number ordp(N)\mathrm{ord}_p(N) ordp(Δ)\mathrm{ord}_p(\Delta) ordp(den(j))\mathrm{ord}_p(\mathrm{den}(j))
22 33 I3I_{3} split multiplicative -1 1 3 3
1313 22 IIIIII additive -1 2 3 0

Copy content comment:Local data
 
Copy content sage:E.local_data()
 
Copy content gp:ellglobalred(E)[5]
 
Copy content magma:[LocalInformation(E,p) : p in BadPrimes(E)];
 
Copy content oscar:[(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
 

Galois representations

The \ell-adic Galois representation has maximal image for all primes \ell except those listed in the table below.

prime \ell mod-\ell image \ell-adic image
33 3Ns 3.6.0.1
55 5B 5.6.0.1

Copy content comment:Mod p Galois image
 
Copy content sage:rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
Copy content magma:[GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

Copy content comment:Adelic image of Galois representation
 
Copy content sage:gens = [[481, 480, 1080, 481], [521, 0, 0, 521], [1351, 630, 0, 1351], [1227, 1397, 10, 873], [391, 150, 1185, 691], [1261, 30, 1290, 1297], [1, 1248, 240, 1], [1, 0, 1500, 1], [736, 1095, 1425, 151], [781, 0, 1500, 781], [521, 150, 0, 1], [1081, 0, 0, 241], [1, 0, 30, 1], [361, 1410, 750, 181], [16, 375, 765, 1306], [1, 0, 1080, 1]] GL(2,Integers(1560)).subgroup(gens)
 
Copy content magma:Gens := [[481, 480, 1080, 481], [521, 0, 0, 521], [1351, 630, 0, 1351], [1227, 1397, 10, 873], [391, 150, 1185, 691], [1261, 30, 1290, 1297], [1, 1248, 240, 1], [1, 0, 1500, 1], [736, 1095, 1425, 151], [781, 0, 1500, 781], [521, 150, 0, 1], [1081, 0, 0, 241], [1, 0, 30, 1], [361, 1410, 750, 181], [16, 375, 765, 1306], [1, 0, 1080, 1]]; sub<GL(2,Integers(1560))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 1560=233513 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 , index 576576, genus 1717, and generators

(4814801080481),(52100521),(135163001351),(1227139710873),(3911501185691),(12613012901297),(112482401),(1015001),(73610951425151),(78101500781),(52115001),(108100241),(10301),(3611410750181),(163757651306),(1010801)\left(\begin{array}{rr} 481 & 480 \\ 1080 & 481 \end{array}\right),\left(\begin{array}{rr} 521 & 0 \\ 0 & 521 \end{array}\right),\left(\begin{array}{rr} 1351 & 630 \\ 0 & 1351 \end{array}\right),\left(\begin{array}{rr} 1227 & 1397 \\ 10 & 873 \end{array}\right),\left(\begin{array}{rr} 391 & 150 \\ 1185 & 691 \end{array}\right),\left(\begin{array}{rr} 1261 & 30 \\ 1290 & 1297 \end{array}\right),\left(\begin{array}{rr} 1 & 1248 \\ 240 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1500 & 1 \end{array}\right),\left(\begin{array}{rr} 736 & 1095 \\ 1425 & 151 \end{array}\right),\left(\begin{array}{rr} 781 & 0 \\ 1500 & 781 \end{array}\right),\left(\begin{array}{rr} 521 & 150 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1081 & 0 \\ 0 & 241 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 30 & 1 \end{array}\right),\left(\begin{array}{rr} 361 & 1410 \\ 750 & 181 \end{array}\right),\left(\begin{array}{rr} 16 & 375 \\ 765 & 1306 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1080 & 1 \end{array}\right).

Input positive integer mm to see the generators of the reduction of HH to GL2(Z/mZ)\mathrm{GL}_2(\Z/m\Z):

The torsion field K:=Q(E[1560])K:=\Q(E[1560]) is a degree-16102195201610219520 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/1560Z)\GL_2(\Z/1560\Z).

The table below list all primes \ell for which the Serre invariants associated to the mod-\ell Galois representation are exceptional.

\ell Reduction type Serre weight Serre conductor
22 split multiplicative 44 13 13
33 good 22 169=132 169 = 13^{2}
1313 additive 5050 2 2

Isogenies

Copy content comment:Isogenies
 
Copy content gp:ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree dd for d=d= 5.
Its isogeny class 338.d consists of 2 curves linked by isogenies of degree 5.

Twists

This elliptic curve is its own minimal quadratic twist.

Growth of torsion in number fields

The number fields KK of degree less than 24 such that E(K)torsE(K)_{\rm tors} is strictly larger than E(Q)torsE(\Q)_{\rm tors} (which is trivial) are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
33 3.1.104.1 Z/2Z\Z/2\Z not in database
44 4.2.6591.1 Z/3Z\Z/3\Z not in database
44 4.0.19773.1 Z/3Z\Z/3\Z not in database
44 4.0.2197.1 Z/5Z\Z/5\Z not in database
66 6.0.1124864.1 Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
88 8.0.390971529.2 Z/3ZZ/3Z\Z/3\Z \oplus \Z/3\Z not in database
88 8.0.43441281.1 Z/15Z\Z/15\Z not in database
1212 12.2.1423311812421484544.1 Z/4Z\Z/4\Z not in database
1212 deg 12 Z/6Z\Z/6\Z not in database
1212 deg 12 Z/6Z\Z/6\Z not in database
1212 12.0.43436029431808.1 Z/10Z\Z/10\Z not in database
1616 16.0.152858736488597841.1 Z/3ZZ/15Z\Z/3\Z \oplus \Z/15\Z not in database
2020 20.4.102371786028181514000000000000000.1 Z/5Z\Z/5\Z not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.

Iwasawa invariants

pp 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type split ord ord ord ss add ord ord ord ss ss ord ss ord ord
λ\lambda-invariant(s) 2 1 1 1 1,1 - 1 1 1 1,1 1,1 1 1,1 1 1
μ\mu-invariant(s) 0 0 0 0 0,0 - 0 0 0 0,0 0,0 0 0,0 0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

pp-adic regulators

Note: pp-adic regulator data only exists for primes p5p\ge 5 of good ordinary reduction.