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Properties

Label 432960.x2
Conductor $432960$
Discriminant $1.260\times 10^{21}$
j-invariant \( \frac{10158546821506606081}{4808208951562500} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z \oplus \Z/{2}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-x^2-2887681x-805042175\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-x^2z-2887681xz^2-805042175z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-233902188x-587577452112\) Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([0, -1, 0, -2887681, -805042175])
 
Copy content gp:E = ellinit([0, -1, 0, -2887681, -805042175])
 
Copy content magma:E := EllipticCurve([0, -1, 0, -2887681, -805042175]);
 
Copy content oscar:E = elliptic_curve([0, -1, 0, -2887681, -805042175])
 
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

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

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

Mordell-Weil generators

$P$$\hat{h}(P)$Order
\( \left(-881, 32472\right) \)$2.2871280818885692592237803282$$\infty$
\( \left(67425, 17502080\right) \)$5.9204614158018132552389999606$$\infty$
\( \left(-287, 0\right) \)$0$$2$
\( \left(1825, 0\right) \)$0$$2$

$P$$\hat{h}(P)$Order
\([-881:32472:1]\)$2.2871280818885692592237803282$$\infty$
\([67425:17502080:1]\)$5.9204614158018132552389999606$$\infty$
\([-287:0:1]\)$0$$2$
\([1825:0:1]\)$0$$2$

$P$$\hat{h}(P)$Order
\( \left(-7932, 876744\right) \)$2.2871280818885692592237803282$$\infty$
\( \left(606822, 472556160\right) \)$5.9204614158018132552389999606$$\infty$
\( \left(-2586, 0\right) \)$0$$2$
\( \left(16422, 0\right) \)$0$$2$

Integral points

\( \left(-1537, 0\right) \), \((-881,\pm 32472)\), \((-799,\pm 31488)\), \((-576,\pm 25823)\), \( \left(-287, 0\right) \), \( \left(1825, 0\right) \), \((67425,\pm 17502080)\), \((610081,\pm 476517888)\) Copy content Toggle raw display

\([-1537:0:1]\), \([-881:\pm 32472:1]\), \([-799:\pm 31488:1]\), \([-576:\pm 25823:1]\), \([-287:0:1]\), \([1825:0:1]\), \([67425:\pm 17502080:1]\), \([610081:\pm 476517888:1]\) Copy content Toggle raw display

\( \left(-1537, 0\right) \), \((-881,\pm 32472)\), \((-799,\pm 31488)\), \((-576,\pm 25823)\), \( \left(-287, 0\right) \), \( \left(1825, 0\right) \), \((67425,\pm 17502080)\), \((610081,\pm 476517888)\) Copy content Toggle raw display

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

Invariants

Conductor: $N$  =  \( 432960 \) = $2^{6} \cdot 3 \cdot 5 \cdot 11 \cdot 41$
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)
 
Minimal Discriminant: $\Delta$  =  $1260443127398400000000$ = $2^{20} \cdot 3^{2} \cdot 5^{8} \cdot 11^{2} \cdot 41^{4} $
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: $j$  =  \( \frac{10158546821506606081}{4808208951562500} \) = $2^{-2} \cdot 3^{-2} \cdot 5^{-8} \cdot 11^{-2} \cdot 13^{3} \cdot 41^{-4} \cdot 166597^{3}$
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: $h_{\mathrm{Faltings}}$ ≈ $2.7429148570360570737269104490$
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $1.7031940861961391096010622668$
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $0.9697120774642616$
Szpiro ratio: $\sigma_{m}$ ≈ $4.333468965556785$
Intrinsic torsion order: $\#E(\mathbb Q)_\text{tors}^\text{is}$ = $1$

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: $\mathrm{Reg}(E/\Q)$ ≈ $13.399096834061680703170168110$
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: $\Omega$ ≈ $0.12130881941214131667397102980$
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 $  = $ 2^{2}\cdot2\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: $\#E(\Q)_{\mathrm{tor}}$ = $4$
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!$ ≈ $13.003428945031862965511043610 $
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

$$\begin{aligned} 13.003428945 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.121309 \cdot 13.399097 \cdot 128}{4^2} \\ & \approx 13.003428945\end{aligned}$$

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, -2887681, -805042175]); 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, -2887681, -805042175]); 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 432960.2.a.x

\( q - q^{3} - q^{5} + q^{9} - q^{11} + 2 q^{13} + q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 

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

Modular degree: 14155776
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
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 5 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$2$ $4$ $I_{10}^{*}$ additive 1 6 20 2
$3$ $2$ $I_{2}$ nonsplit multiplicative 1 1 2 2
$5$ $2$ $I_{8}$ nonsplit multiplicative 1 1 8 8
$11$ $2$ $I_{2}$ nonsplit multiplicative 1 1 2 2
$41$ $4$ $I_{4}$ split multiplicative -1 1 4 4

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 $\ell$-adic index
$2$ 2Cs 8.48.0.106 $48$

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 = [[10821, 2698, 2714, 21], [1, 2714, 0, 8119], [1, 0, 8, 1], [5, 4, 10820, 10821], [10817, 8, 10816, 9], [7879, 2, 2934, 10819], [10297, 8, 8716, 33], [7223, 2, 10806, 10819], [1, 8, 0, 1]] GL(2,Integers(10824)).subgroup(gens)
 
Copy content magma:Gens := [[10821, 2698, 2714, 21], [1, 2714, 0, 8119], [1, 0, 8, 1], [5, 4, 10820, 10821], [10817, 8, 10816, 9], [7879, 2, 2934, 10819], [10297, 8, 8716, 33], [7223, 2, 10806, 10819], [1, 8, 0, 1]]; sub<GL(2,Integers(10824))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10824 = 2^{3} \cdot 3 \cdot 11 \cdot 41 \), index $192$, genus $1$, and generators

$\left(\begin{array}{rr} 10821 & 2698 \\ 2714 & 21 \end{array}\right),\left(\begin{array}{rr} 1 & 2714 \\ 0 & 8119 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 10820 & 10821 \end{array}\right),\left(\begin{array}{rr} 10817 & 8 \\ 10816 & 9 \end{array}\right),\left(\begin{array}{rr} 7879 & 2 \\ 2934 & 10819 \end{array}\right),\left(\begin{array}{rr} 10297 & 8 \\ 8716 & 33 \end{array}\right),\left(\begin{array}{rr} 7223 & 2 \\ 10806 & 10819 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.

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

The torsion field $K:=\Q(E[10824])$ is a degree-$13965557760000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10824\Z)$.

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
$2$ additive $2$ \( 1 \)
$3$ nonsplit multiplicative $4$ \( 144320 = 2^{6} \cdot 5 \cdot 11 \cdot 41 \)
$5$ nonsplit multiplicative $6$ \( 86592 = 2^{6} \cdot 3 \cdot 11 \cdot 41 \)
$11$ nonsplit multiplicative $12$ \( 39360 = 2^{6} \cdot 3 \cdot 5 \cdot 41 \)
$41$ split multiplicative $42$ \( 10560 = 2^{6} \cdot 3 \cdot 5 \cdot 11 \)

Isogenies

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

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 432960.x consists of 6 curves linked by isogenies of degrees dividing 8.

Twists

The minimal quadratic twist of this elliptic curve is 13530.w2, its twist by $8$.

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.