Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-891424202x+13409666278692\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-891424202xz^2+13409666278692z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1155285765819x+625644855755951382\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 454854 \) | = | $2 \cdot 3 \cdot 41 \cdot 43^{2}$ |
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| Discriminant: | $\Delta$ | = | $-32347773351461126079149801472$ | = | $-1 \cdot 2^{15} \cdot 3^{6} \cdot 41^{5} \cdot 43^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{6702384370794891073}{2767558099894272} \) | = | $-1 \cdot 2^{-15} \cdot 3^{-6} \cdot 41^{-5} \cdot 43 \cdot 307^{3} \cdot 1753^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.1787447882363004553997858520$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.6712780444405921730845575098$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9883194673941648$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.678436721912165$ | |||
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: | $\Omega$ | ≈ | $0.034656057279043763121738647206$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 450 $ = $ ( 3 \cdot 5 )\cdot( 2 \cdot 3 )\cdot5\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $15.595225775569693404782391243 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 15.595225776 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.034656 \cdot 1.000000 \cdot 450}{1^2} \\ & \approx 15.595225776\end{aligned}$$
Modular invariants
Modular form 454854.2.a.y
For more coefficients, see the Downloads section to the right.
| Modular degree: | 591645600 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $15$ | $I_{15}$ | split multiplicative | -1 | 1 | 15 | 15 |
| $3$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $41$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $43$ | $1$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
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 \( 328 = 2^{3} \cdot 41 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 247 & 2 \\ 247 & 3 \end{array}\right),\left(\begin{array}{rr} 129 & 2 \\ 129 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 327 & 0 \end{array}\right),\left(\begin{array}{rr} 165 & 2 \\ 165 & 3 \end{array}\right),\left(\begin{array}{rr} 327 & 2 \\ 326 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[328])$ is a degree-$2115993600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/328\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 75809 = 41 \cdot 43^{2} \) |
| $3$ | split multiplicative | $4$ | \( 75809 = 41 \cdot 43^{2} \) |
| $5$ | good | $2$ | \( 5547 = 3 \cdot 43^{2} \) |
| $41$ | split multiplicative | $42$ | \( 11094 = 2 \cdot 3 \cdot 43^{2} \) |
| $43$ | additive | $674$ | \( 246 = 2 \cdot 3 \cdot 41 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 454854y consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 454854b1, 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 $0$.