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Properties

Label 489762dg6
Conductor $489762$
Discriminant $-6.250\times 10^{23}$
j-invariant \( \frac{27207619911317663}{177609314617308} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3-x^2+9530554x+36307705425\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-x^2z+9530554xz^2+36307705425z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+152488869x+2323845636086\) Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([1, -1, 1, 9530554, 36307705425])
 
Copy content gp:E = ellinit([1, -1, 1, 9530554, 36307705425])
 
Copy content magma:E := EllipticCurve([1, -1, 1, 9530554, 36307705425]);
 
Copy content oscar:E = elliptic_curve([1, -1, 1, 9530554, 36307705425])
 
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
\( \left(-1195, 152949\right) \)$2.4008303503761945281363232703$$\infty$
\( \left(-\frac{33939}{16}, \frac{5245161}{64}\right) \)$8.7544062036824032172385394859$$\infty$
\( \left(-\frac{9541}{4}, \frac{9537}{8}\right) \)$0$$2$

$P$$\hat{h}(P)$Order
\([-1195:152949:1]\)$2.4008303503761945281363232703$$\infty$
\([-135756:5245161:64]\)$8.7544062036824032172385394859$$\infty$
\([-19082:9537:8]\)$0$$2$

$P$$\hat{h}(P)$Order
\( \left(-4781, 1218816\right) \)$2.4008303503761945281363232703$$\infty$
\( \left(-\frac{33943}{4}, \frac{5177315}{8}\right) \)$8.7544062036824032172385394859$$\infty$
\( \left(-9542, 0\right) \)$0$$2$

Integral points

\( \left(-1195, 152949\right) \), \( \left(-1195, -151755\right) \), \( \left(19965, 2850849\right) \), \( \left(19965, -2870815\right) \) Copy content Toggle raw display

\([-1195:152949:1]\), \([-1195:-151755:1]\), \([19965:2850849:1]\), \([19965:-2870815:1]\) Copy content Toggle raw display

\((-4781,\pm 1218816)\), \((79859,\pm 22886656)\) Copy content Toggle raw display

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

Invariants

Conductor: $N$  =  \( 489762 \) = $2 \cdot 3^{2} \cdot 7 \cdot 13^{2} \cdot 23$
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: $\Delta$  =  $-624961667705138627615388$ = $-1 \cdot 2^{2} \cdot 3^{10} \cdot 7 \cdot 13^{6} \cdot 23^{8} $
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: $j$  =  \( \frac{27207619911317663}{177609314617308} \) = $2^{-2} \cdot 3^{-4} \cdot 7^{-1} \cdot 23^{-8} \cdot 167^{3} \cdot 1801^{3}$
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: $h_{\mathrm{Faltings}}$ ≈ $3.2477553761046609251493579213$
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $1.4159745530398377114249915821$
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $1.0174819765013208$
Szpiro ratio: $\sigma_{m}$ ≈ $4.743963592629722$
Intrinsic torsion order: $\#E(\mathbb Q)_\text{tors}^\text{is}$ = $2$

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: $\mathrm{Reg}(E/\Q)$ ≈ $20.025578432173558414700251344$
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: $\Omega$ ≈ $0.066265697365888281326097532319$
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: $\prod_{p}c_p$ = $ 64 $  = $ 2\cdot2\cdot1\cdot2\cdot2^{3} $
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!$ ≈ $21.232142719412360774868436694 $
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

$$\begin{aligned} 21.232142719 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.066266 \cdot 20.025578 \cdot 64}{2^2} \\ & \approx 21.232142719\end{aligned}$$

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, 9530554, 36307705425]); 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, 9530554, 36307705425]); 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 489762.2.a.dg

\( q + q^{2} + q^{4} - 2 q^{5} - q^{7} + q^{8} - 2 q^{10} - 4 q^{11} - q^{14} + q^{16} + 6 q^{17} - 4 q^{19} + O(q^{20}) \) 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: 75497472
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 489762dg1 is optimal.

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 $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$2$ $2$ $I_{2}$ split multiplicative -1 1 2 2
$3$ $2$ $I_{4}^{*}$ additive -1 2 10 4
$7$ $1$ $I_{1}$ nonsplit multiplicative 1 1 1 1
$13$ $2$ $I_0^{*}$ additive 1 2 6 0
$23$ $8$ $I_{8}$ split multiplicative -1 1 8 8

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
$2$ 2B 8.24.0.90 $24$

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 = [[100449, 16, 100448, 17], [55576, 48945, 77727, 38650], [15, 2, 100366, 100451], [1, 16, 0, 1], [85021, 79872, 2964, 80185], [5, 4, 100460, 100461], [22231, 79872, 74646, 60217], [1, 0, 16, 1], [91729, 79872, 20280, 36193], [33487, 0, 0, 100463], [15455, 0, 0, 100463]] GL(2,Integers(100464)).subgroup(gens)
 
Copy content magma:Gens := [[100449, 16, 100448, 17], [55576, 48945, 77727, 38650], [15, 2, 100366, 100451], [1, 16, 0, 1], [85021, 79872, 2964, 80185], [5, 4, 100460, 100461], [22231, 79872, 74646, 60217], [1, 0, 16, 1], [91729, 79872, 20280, 36193], [33487, 0, 0, 100463], [15455, 0, 0, 100463]]; sub<GL(2,Integers(100464))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 100464 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \cdot 23 \), index $192$, genus $1$, and generators

$\left(\begin{array}{rr} 100449 & 16 \\ 100448 & 17 \end{array}\right),\left(\begin{array}{rr} 55576 & 48945 \\ 77727 & 38650 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 100366 & 100451 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 85021 & 79872 \\ 2964 & 80185 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 100460 & 100461 \end{array}\right),\left(\begin{array}{rr} 22231 & 79872 \\ 74646 & 60217 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 91729 & 79872 \\ 20280 & 36193 \end{array}\right),\left(\begin{array}{rr} 33487 & 0 \\ 0 & 100463 \end{array}\right),\left(\begin{array}{rr} 15455 & 0 \\ 0 & 100463 \end{array}\right)$.

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

The torsion field $K:=\Q(E[100464])$ is a degree-$86728144349822976$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/100464\Z)$.

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
$2$ split multiplicative $4$ \( 10647 = 3^{2} \cdot 7 \cdot 13^{2} \)
$3$ additive $8$ \( 54418 = 2 \cdot 7 \cdot 13^{2} \cdot 23 \)
$7$ nonsplit multiplicative $8$ \( 69966 = 2 \cdot 3^{2} \cdot 13^{2} \cdot 23 \)
$13$ additive $86$ \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
$23$ split multiplicative $24$ \( 21294 = 2 \cdot 3^{2} \cdot 7 \cdot 13^{2} \)

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 489762dg consists of 6 curves linked by isogenies of degrees dividing 8.

Twists

The minimal quadratic twist of this elliptic curve is 966g6, its twist by $-39$.

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.