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Properties

Label 431970.y6
Conductor $431970$
Discriminant $-1.802\times 10^{24}$
j-invariant \( \frac{8854313460877886399}{1016927675429790600} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3+x^2+5215098x+64416771324\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3+x^2z+5215098xz^2+64416771324z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+6758766333x+3005327501394174\) Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([1, 1, 0, 5215098, 64416771324])
 
Copy content gp:E = ellinit([1, 1, 0, 5215098, 64416771324])
 
Copy content magma:E := EllipticCurve([1, 1, 0, 5215098, 64416771324]);
 
Copy content oscar:E = elliptic_curve([1, 1, 0, 5215098, 64416771324])
 
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

\(\Z \oplus \Z \oplus \Z/{2}\Z\)

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

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$(-2977, 151551)$$2.1392775145174901202949570427$$\infty$
$(9763, 1017871)$$2.8510682724077362220223272695$$\infty$
$(-14309/4, 14309/8)$$0$$2$

Integral points

\( \left(-2977, 151551\right) \), \( \left(-2977, -148574\right) \), \( \left(-2095, 211527\right) \), \( \left(-2095, -209432\right) \), \( \left(9763, 1017871\right) \), \( \left(9763, -1027634\right) \), \( \left(25835, 4163622\right) \), \( \left(25835, -4189457\right) \), \( \left(69053, 18123036\right) \), \( \left(69053, -18192089\right) \) Copy content Toggle raw display

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

Invariants

Conductor: $N$  =  \( 431970 \) = $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2} \cdot 17$
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: $\Delta$  =  $-1801549409612075265126600$ = $-1 \cdot 2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7^{16} \cdot 11^{6} \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: $j$  =  \( \frac{8854313460877886399}{1016927675429790600} \) = $2^{-3} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-16} \cdot 17^{-1} \cdot 47^{3} \cdot 44017^{3}$
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: $\mathrm{End}(E)$ = $\Z$
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$  =  \(\Z\)    (no potential complex multiplication)
Copy content comment:Potential complex multiplication
 
Copy content sage:E.has_cm()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$
Faltings height: $h_{\mathrm{Faltings}}$ ≈ $3.3336065534892358940350210862$
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $2.1346589170900506220040492972$
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $1.036853983278124$
Szpiro ratio: $\sigma_{m}$ ≈ $4.87822243051925$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$ = $ 2$
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: $r$ = $ 2$
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: $\mathrm{Reg}(E/\Q)$ ≈ $5.0709354673288078889570553779$
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: $\Omega$ ≈ $0.064208284524429321463031132137$
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: $\prod_{p}c_p$ = $ 128 $  = $ 1\cdot2\cdot2\cdot2^{4}\cdot2\cdot1 $
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: $\#E(\Q)_{\mathrm{tor}}$ = $2$
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^{(2)}(E,1)/2!$ ≈ $10.419074153320578068149135068 $
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: Ш${}_{\mathrm{an}}$  ≈  $1$    (rounded)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

$$\begin{aligned} 10.419074153 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.064208 \cdot 5.070935 \cdot 128}{2^2} \\ & \approx 10.419074153\end{aligned}$$

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, 0, 5215098, 64416771324]); 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, 0, 5215098, 64416771324]); 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 431970.2.a.y

\( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - q^{10} - q^{12} - 6 q^{13} - q^{14} - q^{15} + q^{16} - q^{17} - q^{18} - 4 q^{19} + O(q^{20}) \) 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: 94371840
Copy content comment:Modular degree
 
Copy content sage:E.modular_degree()
 
Copy content gp:ellmoddegree(E)
 
Copy content magma:ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: no
Manin constant: 1 (conditional*)
Copy content comment:Manin constant
 
Copy content magma:ManinConstant(E);
 
* The Manin constant is correct provided that curve 431970.y5 is optimal.

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 6 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$2$ $1$ $I_{3}$ nonsplit multiplicative 1 1 3 3
$3$ $2$ $I_{2}$ nonsplit multiplicative 1 1 2 2
$5$ $2$ $I_{2}$ split multiplicative -1 1 2 2
$7$ $16$ $I_{16}$ split multiplicative -1 1 16 16
$11$ $2$ $I_0^{*}$ additive -1 2 6 0
$17$ $1$ $I_{1}$ 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
$2$ 2B 16.24.0.1

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 = [[10088, 4081, 10879, 36730], [1, 16, 0, 1], [22958, 15301, 13849, 14290], [27556, 15301, 42999, 14290], [4079, 0, 0, 44879], [31021, 20416, 23408, 11925], [1, 0, 16, 1], [15, 2, 44782, 44867], [34013, 20416, 2464, 11925], [5, 4, 44876, 44877], [44865, 16, 44864, 17]] GL(2,Integers(44880)).subgroup(gens)
 
Copy content magma:Gens := [[10088, 4081, 10879, 36730], [1, 16, 0, 1], [22958, 15301, 13849, 14290], [27556, 15301, 42999, 14290], [4079, 0, 0, 44879], [31021, 20416, 23408, 11925], [1, 0, 16, 1], [15, 2, 44782, 44867], [34013, 20416, 2464, 11925], [5, 4, 44876, 44877], [44865, 16, 44864, 17]]; sub<GL(2,Integers(44880))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 44880 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 17 \), index $192$, genus $1$, and generators

$\left(\begin{array}{rr} 10088 & 4081 \\ 10879 & 36730 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 22958 & 15301 \\ 13849 & 14290 \end{array}\right),\left(\begin{array}{rr} 27556 & 15301 \\ 42999 & 14290 \end{array}\right),\left(\begin{array}{rr} 4079 & 0 \\ 0 & 44879 \end{array}\right),\left(\begin{array}{rr} 31021 & 20416 \\ 23408 & 11925 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 44782 & 44867 \end{array}\right),\left(\begin{array}{rr} 34013 & 20416 \\ 2464 & 11925 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 44876 & 44877 \end{array}\right),\left(\begin{array}{rr} 44865 & 16 \\ 44864 & 17 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[44880])$ is a degree-$3049493889024000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/44880\Z)$.

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
$2$ nonsplit multiplicative $4$ \( 2057 = 11^{2} \cdot 17 \)
$3$ nonsplit multiplicative $4$ \( 71995 = 5 \cdot 7 \cdot 11^{2} \cdot 17 \)
$5$ split multiplicative $6$ \( 86394 = 2 \cdot 3 \cdot 7 \cdot 11^{2} \cdot 17 \)
$7$ split multiplicative $8$ \( 61710 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \cdot 17 \)
$11$ additive $62$ \( 3570 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \)
$17$ nonsplit multiplicative $18$ \( 25410 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2} \)

Isogenies

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

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 431970.y consists of 6 curves linked by isogenies of degrees dividing 8.

Twists

The minimal quadratic twist of this elliptic curve is 3570.t6, its twist by $-11$.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

$p$-adic regulators

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.