Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+998250x+397594656\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+998250xz^2+397594656z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+15972000x+25446058000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-2420/9, 282824/27)$ | $2.1813797167967718983392224001$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 462825 \) | = | $3^{2} \cdot 5^{2} \cdot 11^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-131955800270136046875$ | = | $-1 \cdot 3^{6} \cdot 5^{6} \cdot 11^{9} \cdot 17^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{4096000}{4913} \) | = | $2^{15} \cdot 5^{3} \cdot 17^{-3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5469531518370365891464242483$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.60549340331284635189803572025$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9072219820005657$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.072411498929203$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.1813797167967718983392224001$ |
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| Real period: | $\Omega$ | ≈ | $0.12362632030085489220645865995$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 2\cdot2\cdot2\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.4722227415961405735041673342 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.472222742 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.123626 \cdot 2.181380 \cdot 24}{1^2} \\ & \approx 6.472222742\end{aligned}$$
Modular invariants
Modular form 462825.2.a.cj
For more coefficients, see the Downloads section to the right.
| Modular degree: | 13685760 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $11$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $17$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 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 | $\ell$-adic index |
|---|---|---|---|
| $3$ | 3Ns | 9.18.0.1 | $18$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3366 = 2 \cdot 3^{2} \cdot 11 \cdot 17 \), index $72$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1507 & 3002 \\ 1596 & 839 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 2179 & 18 \\ 2781 & 163 \end{array}\right),\left(\begin{array}{rr} 3349 & 18 \\ 3348 & 19 \end{array}\right),\left(\begin{array}{rr} 11 & 10 \\ 1531 & 1 \end{array}\right),\left(\begin{array}{rr} 17 & 18 \\ 84 & 89 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 6 \\ 3186 & 3283 \end{array}\right)$.
The torsion field $K:=\Q(E[3366])$ is a degree-$335027404800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3366\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 42075 = 3^{2} \cdot 5^{2} \cdot 11 \cdot 17 \) |
| $3$ | additive | $2$ | \( 3025 = 5^{2} \cdot 11^{2} \) |
| $5$ | additive | $14$ | \( 18513 = 3^{2} \cdot 11^{2} \cdot 17 \) |
| $11$ | additive | $42$ | \( 3825 = 3^{2} \cdot 5^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 27225 = 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 462825cj consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 2057a1, its twist by $165$.
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.