Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-5195981388x-155507383686512\)
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(homogenize, simplify) |
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\(y^2z=x^3-5195981388xz^2-155507383686512z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5195981388x-155507383686512\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(171986589178362622604692772677069206206041638906524/1048137399281898644895288939303275411847972721, 1981527224729185980221287864555765212164258573623594972498337755056583845360/33933396085560227220023462601984731196863346354477108768068029765769)$ | $113.30420939793087497677665125$ | $\infty$ |
| $(83954, 0)$ | $0$ | $2$ |
Integral points
\( \left(83954, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 486720 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-1468795264900377098560575897600$ | = | $-1 \cdot 2^{48} \cdot 3^{9} \cdot 5^{2} \cdot 13^{9} $ |
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| j-invariant: | $j$ | = | \( -\frac{16818951115904497561}{1592332281446400} \) | = | $-1 \cdot 2^{-30} \cdot 3^{-3} \cdot 5^{-2} \cdot 13^{-3} \cdot 17^{3} \cdot 29^{3} \cdot 5197^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $4.5303140572057098278548173785$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.6588124633009686500046028571$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0346538273212997$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.023359915282825$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $113.30420939793087497677665125$ |
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| Real period: | $\Omega$ | ≈ | $0.0088434445484546870656968438889$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.0159959433368009383719284451 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.015995943 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.008843 \cdot 113.304209 \cdot 32}{2^2} \\ & \approx 8.015995943\end{aligned}$$
Modular invariants
Modular form 486720.2.a.gb
For more coefficients, see the Downloads section to the right.
| Modular degree: | 557383680 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{38}^{*}$ | additive | -1 | 6 | 48 | 30 |
| $3$ | $2$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
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 | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 11 & 2 \\ 1510 & 1551 \end{array}\right),\left(\begin{array}{rr} 1549 & 12 \\ 1548 & 13 \end{array}\right),\left(\begin{array}{rr} 937 & 12 \\ 942 & 73 \end{array}\right),\left(\begin{array}{rr} 250 & 1557 \\ 1067 & 8 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 110 & 1557 \\ 1107 & 8 \end{array}\right),\left(\begin{array}{rr} 1485 & 1232 \\ 1522 & 1243 \end{array}\right),\left(\begin{array}{rr} 779 & 1548 \\ 1554 & 1487 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$9661317120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $2$ | \( 1521 = 3^{2} \cdot 13^{2} \) |
| $3$ | additive | $6$ | \( 54080 = 2^{6} \cdot 5 \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 97344 = 2^{6} \cdot 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 486720gb
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 390d3, its twist by $312$.
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.