Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-838071307x+9385479533753\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-838071307xz^2+9385479533753z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1086140413899x+437905225232996742\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(355699/25, 72203358/125)$ | $9.5862545345079096963540170534$ | $\infty$ |
| $(-133789/4, 133785/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 454854 \) | = | $2 \cdot 3 \cdot 41 \cdot 43^{2}$ |
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| Discriminant: | $\Delta$ | = | $-384147958768120049541950592$ | = | $-1 \cdot 2^{7} \cdot 3^{24} \cdot 41^{2} \cdot 43^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{10298071306410575356297}{60769798505543808} \) | = | $-1 \cdot 2^{-7} \cdot 3^{-24} \cdot 41^{-2} \cdot 21756313^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.9415535382236012400616291493$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.0609534803768200283252078926$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0617280385727412$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.62366427384615$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $9.5862545345079096963540170534$ |
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| Real period: | $\Omega$ | ≈ | $0.053770487438420211628616051725$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 112 $ = $ 7\cdot2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $14.432812212819177841152038142 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.432812213 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.053770 \cdot 9.586255 \cdot 112}{2^2} \\ & \approx 14.432812213\end{aligned}$$
Modular invariants
Modular form 454854.2.a.q
For more coefficients, see the Downloads section to the right.
| Modular degree: | 252887040 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $3$ | $2$ | $I_{24}$ | nonsplit multiplicative | 1 | 1 | 24 | 24 |
| $41$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $43$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 | 8.6.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 328 = 2^{3} \cdot 41 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 325 & 4 \\ 324 & 5 \end{array}\right),\left(\begin{array}{rr} 124 & 209 \\ 41 & 288 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 129 & 4 \\ 258 & 9 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 163 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[328])$ is a degree-$352665600$ 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 | $4$ | \( 1849 = 43^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 151618 = 2 \cdot 41 \cdot 43^{2} \) |
| $7$ | good | $2$ | \( 227427 = 3 \cdot 41 \cdot 43^{2} \) |
| $41$ | nonsplit multiplicative | $42$ | \( 11094 = 2 \cdot 3 \cdot 43^{2} \) |
| $43$ | additive | $926$ | \( 246 = 2 \cdot 3 \cdot 41 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 454854.q
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 246.c2, its twist by $-43$.
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.