Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-2812x-190093\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-2812xz^2-190093z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3644379x-8814305034\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2263/4, 104975/8)$ | $4.0321140893316620431772539247$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 454854 \) | = | $2 \cdot 3 \cdot 41 \cdot 43^{2}$ |
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| Discriminant: | $\Delta$ | = | $-13995497790486$ | = | $-1 \cdot 2 \cdot 3^{3} \cdot 41 \cdot 43^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{389017}{2214} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-3} \cdot 41^{-1} \cdot 73^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2063948564404568910184458618$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.67420520140632432071797539487$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8755235934443263$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.903138950333266$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.0321140893316620431772539247$ |
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| Real period: | $\Omega$ | ≈ | $0.29402952255087619218155124402$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.3711211611136990919871209462 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.371121161 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.294030 \cdot 4.032114 \cdot 2}{1^2} \\ & \approx 2.371121161\end{aligned}$$
Modular invariants
Modular form 454854.2.a.m
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1572480 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $41$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $43$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
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 $16$, genus $0$, and generators
$\left(\begin{array}{rr} 12342 & 14233 \\ 6235 & 39174 \end{array}\right),\left(\begin{array}{rr} 8855 & 0 \\ 0 & 42311 \end{array}\right),\left(\begin{array}{rr} 31735 & 26574 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 42307 & 6 \\ 42306 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21157 & 26574 \\ 13287 & 37411 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 30961 & 26574 \\ 387 & 37411 \end{array}\right)$.
The torsion field $K:=\Q(E[42312])$ is a degree-$42372619488460800$ 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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 227427 = 3 \cdot 41 \cdot 43^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 151618 = 2 \cdot 41 \cdot 43^{2} \) |
| $41$ | 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 $d=$
3.
Its isogeny class 454854.m
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 246.d1, 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.