Properties

Label 4080r1
Conductor 40804080
Discriminant 36720-36720
j-invariant 17559042295 -\frac{1755904}{2295}
CM no
Rank 00
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2=x3x26x9y^2=x^3-x^2-6x-9 Copy content Toggle raw display (homogenize, simplify)
y2z=x3x2z6xz29z3y^2z=x^3-x^2z-6xz^2-9z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x3513x8073y^2=x^3-513x-8073 Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([0, -1, 0, -6, -9])
 
Copy content gp:E = ellinit([0, -1, 0, -6, -9])
 
Copy content magma:E := EllipticCurve([0, -1, 0, -6, -9]);
 
Copy content oscar:E = elliptic_curve([0, -1, 0, -6, -9])
 
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

trivial

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

Invariants

Conductor: NN  =  4080 4080  = 2435172^{4} \cdot 3 \cdot 5 \cdot 17
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  =  36720-36720 = 12433517-1 \cdot 2^{4} \cdot 3^{3} \cdot 5 \cdot 17
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  =  17559042295 -\frac{1755904}{2295}  = 1283351171193-1 \cdot 2^{8} \cdot 3^{-3} \cdot 5^{-1} \cdot 17^{-1} \cdot 19^{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}} ≈ 0.43111733578197039954834887781-0.43111733578197039954834887781
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}} ≈ 0.66216639596861883602075958496-0.66216639596861883602075958496
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
abcabc quality: QQ ≈ 0.7662250439963590.766225043996359
Szpiro ratio: σm\sigma_{m} ≈ 2.20503785282173142.2050378528217314

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 0 0
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 = 0 0
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) = 11
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 ≈ 1.43489387910559795223041968161.4348938791055979522304196816
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 = 1 1
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) ≈ 1.43489387910559795223041968161.4348938791055979522304196816
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    (exact)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

1.434893879L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor211.4348941.0000001121.434893879\begin{aligned} 1.434893879 \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 1.434894 \cdot 1.000000 \cdot 1}{1^2} \\ & \approx 1.434893879\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([0, -1, 0, -6, -9]); 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([0, -1, 0, -6, -9]); 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   4080.2.a.f

qq3q5+q7+q9+3q114q13+q15q17+7q19+O(q20) q - q^{3} - q^{5} + q^{7} + q^{9} + 3 q^{11} - 4 q^{13} + q^{15} - q^{17} + 7 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: 288
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 4 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 11 IIII additive -1 4 4 0
33 11 I3I_{3} nonsplit multiplicative 1 1 3 3
55 11 I1I_{1} nonsplit multiplicative 1 1 1 1
1717 11 I1I_{1} nonsplit multiplicative 1 1 1 1

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 3B 3.4.0.1

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 = [[509, 0, 0, 1019], [4, 3, 9, 7], [594, 929, 677, 576], [241, 6, 213, 19], [1015, 6, 1014, 7], [713, 1014, 609, 1001], [515, 1014, 516, 1013], [1, 6, 0, 1], [3, 4, 8, 11], [1, 0, 6, 1]] GL(2,Integers(1020)).subgroup(gens)
 
Copy content magma:Gens := [[509, 0, 0, 1019], [4, 3, 9, 7], [594, 929, 677, 576], [241, 6, 213, 19], [1015, 6, 1014, 7], [713, 1014, 609, 1001], [515, 1014, 516, 1013], [1, 6, 0, 1], [3, 4, 8, 11], [1, 0, 6, 1]]; sub<GL(2,Integers(1020))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 1020=223517 1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17 , index 1616, genus 00, and generators

(509001019),(4397),(594929677576),(241621319),(1015610147),(71310146091001),(51510145161013),(1601),(34811),(1061)\left(\begin{array}{rr} 509 & 0 \\ 0 & 1019 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 594 & 929 \\ 677 & 576 \end{array}\right),\left(\begin{array}{rr} 241 & 6 \\ 213 & 19 \end{array}\right),\left(\begin{array}{rr} 1015 & 6 \\ 1014 & 7 \end{array}\right),\left(\begin{array}{rr} 713 & 1014 \\ 609 & 1001 \end{array}\right),\left(\begin{array}{rr} 515 & 1014 \\ 516 & 1013 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 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[1020])K:=\Q(E[1020]) is a degree-1082916864010829168640 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/1020Z)\GL_2(\Z/1020\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 additive 22 255=3517 255 = 3 \cdot 5 \cdot 17
33 nonsplit multiplicative 44 1360=24517 1360 = 2^{4} \cdot 5 \cdot 17
55 nonsplit multiplicative 66 816=24317 816 = 2^{4} \cdot 3 \cdot 17
1717 nonsplit multiplicative 1818 240=2435 240 = 2^{4} \cdot 3 \cdot 5

Isogenies

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

This curve has non-trivial cyclic isogenies of degree dd for d=d= 3.
Its isogeny class 4080r consists of 2 curves linked by isogenies of degree 3.

Twists

The minimal quadratic twist of this elliptic curve is 1020e1, its twist by 4-4.

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
22 Q(1)\Q(\sqrt{-1}) Z/3Z\Z/3\Z not in database
33 3.1.255.1 Z/2Z\Z/2\Z not in database
66 6.0.16581375.1 Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
66 6.2.360810720000.7 Z/3Z\Z/3\Z not in database
66 6.0.4161600.2 Z/6Z\Z/6\Z not in database
1212 deg 12 Z/4Z\Z/4\Z not in database
1212 deg 12 Z/3ZZ/3Z\Z/3\Z \oplus \Z/3\Z not in database
1212 deg 12 Z/2ZZ/6Z\Z/2\Z \oplus \Z/6\Z not in database
1818 18.0.9412585696429091573158438502400000000.1 Z/9Z\Z/9\Z not in database
1818 18.2.339372109841273700915916800000000000000.1 Z/6Z\Z/6\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 add nonsplit nonsplit ord ord ord nonsplit ord ss ord ord ord ord ord ord
λ\lambda-invariant(s) - 0 0 2 0 0 0 0 0,0 0 0 0 0 0 0
μ\mu-invariant(s) - 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

All pp-adic regulators are identically 11 since the rank is 00.