Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-1828699x-952573475\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-1828699xz^2-952573475z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2369994579x-44407718134290\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-778, 389)$ | $0$ | $2$ |
Integral points
\( \left(-778, 389\right) \)
Invariants
| Conductor: | $N$ | = | \( 454854 \) | = | $2 \cdot 3 \cdot 41 \cdot 43^{2}$ |
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| Discriminant: | $\Delta$ | = | $19257804959708736$ | = | $2^{6} \cdot 3^{3} \cdot 41 \cdot 43^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{106989222612817}{3046464} \) | = | $2^{-6} \cdot 3^{-3} \cdot 29^{3} \cdot 41^{-1} \cdot 43^{-1} \cdot 1637^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2261447110872457047447926720$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.34554465324046449300837141533$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8927066582455677$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.211857512593241$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.12982690649870750345961014968$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot1\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.0386152519896600276768811974 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 1.038615252 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.129827 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 1.038615252\end{aligned}$$
Modular invariants
Modular form 454854.2.a.e
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6918912 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $41$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $43$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 | 8.6.0.4 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 42312 = 2^{3} \cdot 3 \cdot 41 \cdot 43 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 19610 & 1 \\ 11351 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 28210 & 1 \\ 28207 & 0 \end{array}\right),\left(\begin{array}{rr} 24602 & 1 \\ 8855 & 0 \end{array}\right),\left(\begin{array}{rr} 26449 & 15868 \\ 5288 & 37023 \end{array}\right),\left(\begin{array}{rr} 21157 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 42309 & 4 \\ 42308 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[42312])$ is a degree-$56496825984614400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/42312\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 227427 = 3 \cdot 41 \cdot 43^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 75809 = 41 \cdot 43^{2} \) |
| $41$ | nonsplit multiplicative | $42$ | \( 11094 = 2 \cdot 3 \cdot 43^{2} \) |
| $43$ | additive | $968$ | \( 246 = 2 \cdot 3 \cdot 41 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 454854e
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 10578h1, its twist by $-43$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.