Properties

Label 437320d2
Conductor 437320437320
Discriminant 2.573×1015-2.573\times 10^{15}
j-invariant 44225 -\frac{4}{4225}
CM no
Rank 11
Torsion structure Z/2Z\Z/{2}\Z

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2=x3+x2280x2440800y^2=x^3+x^2-280x-2440800 Copy content Toggle raw display (homogenize, simplify)
y2z=x3+x2z280xz22440800z3y^2z=x^3+x^2z-280xz^2-2440800z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x322707x1779275106y^2=x^3-22707x-1779275106 Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([0, 1, 0, -280, -2440800])
 
Copy content gp:E = ellinit([0, 1, 0, -280, -2440800])
 
Copy content magma:E := EllipticCurve([0, 1, 0, -280, -2440800]);
 
Copy content oscar:E = elliptic_curve([0, 1, 0, -280, -2440800])
 
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

ZZ/2Z\Z \oplus \Z/{2}\Z

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

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(148,884)(148, 884)3.18307326735126512519854370983.1830732673512651251985437098\infty
(135,0)(135, 0)0022

Integral points

(135,0) \left(135, 0\right) , (148,±884)(148,\pm 884), (4340,±285940)(4340,\pm 285940) Copy content Toggle raw display

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

Invariants

Conductor: NN  =  437320 437320  = 235132922^{3} \cdot 5 \cdot 13 \cdot 29^{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  =  2573443615974400-2573443615974400 = 121052132296-1 \cdot 2^{10} \cdot 5^{2} \cdot 13^{2} \cdot 29^{6}
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  =  44225 -\frac{4}{4225}  = 12252132-1 \cdot 2^{2} \cdot 5^{-2} \cdot 13^{-2}
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}} ≈ 1.63588561073200305606224058291.6358856107320030560622405829
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}} ≈ 0.62538495472785504871042220116-0.62538495472785504871042220116
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
abcabc quality: QQ ≈ 1.09430568887334671.0943056888733467
Szpiro ratio: σm\sigma_{m} ≈ 3.3059248364046553.305924836404655

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) ≈ 3.18307326735126512519854370983.1830732673512651251985437098
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 ≈ 0.208972996469832504313377162210.20897299646983250431337716221
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 = 32 32  = 22222 2\cdot2\cdot2\cdot2^{2}
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}} = 22
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) ≈ 5.32141086929131313772530866285.3214108692913131377253086628
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

5.321410869L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.2089733.18307332225.321410869\begin{aligned} 5.321410869 \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 0.208973 \cdot 3.183073 \cdot 32}{2^2} \\ & \approx 5.321410869\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, -280, -2440800]); 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, -280, -2440800]); 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 437320.2.a.d

q2q3+q5+q92q11+q132q152q172q19+O(q20) q - 2 q^{3} + q^{5} + q^{9} - 2 q^{11} + q^{13} - 2 q^{15} - 2 q^{17} - 2 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: 1548288
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: no
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 22 IIIIII^{*} additive 1 3 10 0
55 22 I2I_{2} split multiplicative -1 1 2 2
1313 22 I2I_{2} split multiplicative -1 1 2 2
2929 44 I0I_0^{*} additive 1 2 6 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
22 2B 4.6.0.5

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 = [[3, 8, 10, 27], [1, 0, 8, 1], [15073, 8, 15072, 9], [1, 8, 0, 1], [7799, 0, 0, 15079], [11311, 2088, 2407, 4177], [7541, 2088, 0, 1], [3017, 1566, 0, 1], [5, 8, 48, 77], [10441, 1566, 0, 1]] GL(2,Integers(15080)).subgroup(gens)
 
Copy content magma:Gens := [[3, 8, 10, 27], [1, 0, 8, 1], [15073, 8, 15072, 9], [1, 8, 0, 1], [7799, 0, 0, 15079], [11311, 2088, 2407, 4177], [7541, 2088, 0, 1], [3017, 1566, 0, 1], [5, 8, 48, 77], [10441, 1566, 0, 1]]; sub<GL(2,Integers(15080))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 15080=2351329 15080 = 2^{3} \cdot 5 \cdot 13 \cdot 29 , index 4848, genus 00, and generators

(381027),(1081),(150738150729),(1801),(77990015079),(11311208824074177),(7541208801),(3017156601),(584877),(10441156601)\left(\begin{array}{rr} 3 & 8 \\ 10 & 27 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 15073 & 8 \\ 15072 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7799 & 0 \\ 0 & 15079 \end{array}\right),\left(\begin{array}{rr} 11311 & 2088 \\ 2407 & 4177 \end{array}\right),\left(\begin{array}{rr} 7541 & 2088 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3017 & 1566 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 10441 & 1566 \\ 0 & 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[15080])K:=\Q(E[15080]) is a degree-274574632550400274574632550400 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/15080Z)\GL_2(\Z/15080\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 22 841=292 841 = 29^{2}
55 split multiplicative 66 87464=2313292 87464 = 2^{3} \cdot 13 \cdot 29^{2}
1313 split multiplicative 1414 33640=235292 33640 = 2^{3} \cdot 5 \cdot 29^{2}
2929 additive 422422 520=23513 520 = 2^{3} \cdot 5 \cdot 13

Isogenies

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

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2.
Its isogeny class 437320d consists of 2 curves linked by isogenies of degree 2.

Twists

The minimal quadratic twist of this elliptic curve is 520b2, its twist by 2929.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

pp-adic regulators

pp-adic regulators are not yet computed for curves that are not Γ0\Gamma_0-optimal.