Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-3719244x-2760769136\)
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(homogenize, simplify) |
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\(y^2z=x^3-3719244xz^2-2760769136z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3719244x-2760769136\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(3842, 199152)$ | $5.5034741463206526907419136092$ | $\infty$ |
Integral points
\((3842,\pm 199152)\)
Invariants
| Conductor: | $N$ | = | \( 404928 \) | = | $2^{6} \cdot 3^{2} \cdot 19 \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $440720058875904$ | = | $2^{17} \cdot 3^{14} \cdot 19 \cdot 37 $ |
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| j-invariant: | $j$ | = | \( \frac{59545681581839906}{4612383} \) | = | $2 \cdot 3^{-8} \cdot 19^{-1} \cdot 37^{-1} \cdot 309937^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2572949817793551433896714046$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.72603033165204444268430328074$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9493197066198518$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.414735117018699$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.5034741463206526907419136092$ |
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| Real period: | $\Omega$ | ≈ | $0.10871410352214862917142583783$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.7864420645965759106420432472 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.786442065 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.108714 \cdot 5.503474 \cdot 8}{1^2} \\ & \approx 4.786442065\end{aligned}$$
Modular invariants
Modular form 404928.2.a.j
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6881280 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{7}^{*}$ | additive | -1 | 6 | 17 | 0 |
| $3$ | $2$ | $I_{8}^{*}$ | additive | -1 | 2 | 14 | 8 |
| $19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $37$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 \( 5624 = 2^{3} \cdot 19 \cdot 37 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1407 & 2 \\ 1407 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 2073 & 2 \\ 2073 & 3 \end{array}\right),\left(\begin{array}{rr} 2813 & 2 \\ 2813 & 3 \end{array}\right),\left(\begin{array}{rr} 4257 & 2 \\ 4257 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 5623 & 0 \end{array}\right),\left(\begin{array}{rr} 5623 & 2 \\ 5622 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[5624])$ is a degree-$172297965404160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5624\Z)$.
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 | $4$ | \( 6327 = 3^{2} \cdot 19 \cdot 37 \) |
| $3$ | additive | $8$ | \( 44992 = 2^{6} \cdot 19 \cdot 37 \) |
| $19$ | split 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 404928j consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 16872d1, its twist by $24$.
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.