Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-1519692x-721099120\)
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(homogenize, simplify) |
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\(y^2z=x^3-1519692xz^2-721099120z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1519692x-721099120\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(6940, 568320)$ | $5.2929560425268170081615547446$ | $\infty$ |
Integral points
\((6940,\pm 568320)\)
Invariants
| Conductor: | $N$ | = | \( 404928 \) | = | $2^{6} \cdot 3^{2} \cdot 19 \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $-13950584742739968$ | = | $-1 \cdot 2^{29} \cdot 3^{3} \cdot 19 \cdot 37^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{54838650358603467}{1971009536} \) | = | $-1 \cdot 2^{-11} \cdot 3^{3} \cdot 19^{-1} \cdot 37^{-3} \cdot 126641^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1866205297702221502908209128$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.87224668676327676331616142138$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9624831260790826$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.206782246876903$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.2929560425268170081615547446$ |
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| Real period: | $\Omega$ | ≈ | $0.067987602747417865021560020704$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2\cdot2\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.3182647133462985684329306124 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.318264713 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.067988 \cdot 5.292956 \cdot 12}{1^2} \\ & \approx 4.318264713\end{aligned}$$
Modular invariants
Modular form 404928.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6285312 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{19}^{*}$ | additive | -1 | 6 | 29 | 11 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $19$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $37$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 16872 = 2^{3} \cdot 3 \cdot 19 \cdot 37 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 8437 & 2 \\ 8437 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 15505 & 2 \\ 15505 & 3 \end{array}\right),\left(\begin{array}{rr} 16871 & 2 \\ 16870 & 3 \end{array}\right),\left(\begin{array}{rr} 13321 & 2 \\ 13321 & 3 \end{array}\right),\left(\begin{array}{rr} 12655 & 2 \\ 12655 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 16871 & 0 \end{array}\right),\left(\begin{array}{rr} 11249 & 2 \\ 11249 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[16872])$ is a degree-$8270302339399680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/16872\Z)$.
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 | $4$ | \( 2109 = 3 \cdot 19 \cdot 37 \) |
| $3$ | additive | $6$ | \( 1216 = 2^{6} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 21312 = 2^{6} \cdot 3^{2} \cdot 37 \) |
| $37$ | split multiplicative | $38$ | \( 10944 = 2^{6} \cdot 3^{2} \cdot 19 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 404928.d consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 12654.i1, 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.