Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-539274x+152425825\)
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(homogenize, simplify) |
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\(y^2z=x^3-539274xz^2+152425825z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-539274x+152425825\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(18768/49, 492745/343)$ | $5.6788123987476044539322668139$ | $\infty$ |
| $(425, 0)$ | $0$ | $2$ |
Integral points
\( \left(425, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 478584 \) | = | $2^{3} \cdot 3^{2} \cdot 17^{2} \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $174836715590736$ | = | $2^{4} \cdot 3^{9} \cdot 17^{6} \cdot 23 $ |
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| j-invariant: | $j$ | = | \( \frac{61604313088}{621} \) | = | $2^{11} \cdot 3^{-3} \cdot 23^{-1} \cdot 311^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8918366696510755308217945212$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.30512520689773579147300611335$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0662343776302834$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9153725208238166$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.6788123987476044539322668139$ |
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| Real period: | $\Omega$ | ≈ | $0.51627856223389329121355834145$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.727396401685679802193540790 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.727396402 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.516279 \cdot 5.678812 \cdot 16}{2^2} \\ & \approx 11.727396402\end{aligned}$$
Modular invariants
Modular form 478584.2.a.bg
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3932160 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | 1 | 3 | 4 | 0 |
| $3$ | $2$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $17$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $23$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9384 = 2^{3} \cdot 3 \cdot 17 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 6835 & 6834 \\ 6154 & 6427 \end{array}\right),\left(\begin{array}{rr} 4489 & 4488 \\ 1462 & 1735 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5356 & 1105 \\ 6647 & 5526 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7904 & 6069 \\ 731 & 8278 \end{array}\right),\left(\begin{array}{rr} 9377 & 8 \\ 9376 & 9 \end{array}\right),\left(\begin{array}{rr} 2207 & 0 \\ 0 & 9383 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 9378 & 9379 \end{array}\right)$.
The torsion field $K:=\Q(E[9384])$ is a degree-$32146748080128$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9384\Z)$.
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 | $2$ | \( 59823 = 3^{2} \cdot 17^{2} \cdot 23 \) |
| $3$ | additive | $2$ | \( 53176 = 2^{3} \cdot 17^{2} \cdot 23 \) |
| $17$ | additive | $146$ | \( 1656 = 2^{3} \cdot 3^{2} \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 20808 = 2^{3} \cdot 3^{2} \cdot 17^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 478584.bg
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 552.c3, its twist by $-51$.
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.