Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-16187109x-25035530751\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-16187109xz^2-25035530751z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-20978493291x-1167994787238810\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-2340, 6717)$ | $0.65152207651771357295269961442$ | $\infty$ |
| $(41806/9, 7553/27)$ | $5.1124579034589131492986813669$ | $\infty$ |
| $(-2254, 1127)$ | $0$ | $2$ |
Integral points
\( \left(-2382, 4071\right) \), \( \left(-2382, -1689\right) \), \( \left(-2340, 6717\right) \), \( \left(-2340, -4377\right) \), \( \left(-2304, 6657\right) \), \( \left(-2304, -4353\right) \), \( \left(-2254, 1127\right) \), \( \left(5142, 163839\right) \), \( \left(5142, -168981\right) \), \( \left(8754, 705639\right) \), \( \left(8754, -714393\right) \), \( \left(16242, 1991079\right) \), \( \left(16242, -2007321\right) \)
Invariants
| Conductor: | $N$ | = | \( 454854 \) | = | $2 \cdot 3 \cdot 41 \cdot 43^{2}$ |
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| Discriminant: | $\Delta$ | = | $709919722034702843904$ | = | $2^{18} \cdot 3^{5} \cdot 41 \cdot 43^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{74202895742358313}{112304848896} \) | = | $2^{-18} \cdot 3^{-5} \cdot 7^{3} \cdot 41^{-1} \cdot 43^{-1} \cdot 173^{3} \cdot 347^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9010760411182980035173478701$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0204759832715167917809266134$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9329685146436659$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.714004046254414$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.2881250760708725622521808255$ |
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| Real period: | $\Omega$ | ≈ | $0.075274296858599212705173486670$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 360 $ = $ ( 2 \cdot 3^{2} )\cdot5\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $22.276017277592668183785505383 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 22.276017278 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.075274 \cdot 3.288125 \cdot 360}{2^2} \\ & \approx 22.276017278\end{aligned}$$
Modular invariants
Modular form 454854.2.a.u
For more coefficients, see the Downloads section to the right.
| Modular degree: | 45239040 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $18$ | $I_{18}$ | split multiplicative | -1 | 1 | 18 | 18 |
| $3$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $41$ | $1$ | $I_{1}$ | 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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 227427 = 3 \cdot 41 \cdot 43^{2} \) |
| $3$ | split multiplicative | $4$ | \( 75809 = 41 \cdot 43^{2} \) |
| $5$ | good | $2$ | \( 151618 = 2 \cdot 41 \cdot 43^{2} \) |
| $41$ | 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 454854u
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 10578b1, 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.