Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2+50845x-183778653\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z+50845xz^2-183778653z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+813525x-11761020250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1439, 52830)$ | $3.5289324684379415842846020197$ | $\infty$ |
| $(539, -270)$ | $0$ | $2$ |
Integral points
\( \left(539, -270\right) \), \( \left(1439, 52830\right) \), \( \left(1439, -54270\right) \), \( \left(1563, 60146\right) \), \( \left(1563, -61710\right) \)
Invariants
| Conductor: | $N$ | = | \( 424350 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 23 \cdot 41$ |
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| Discriminant: | $\Delta$ | = | $-14597018615808000000$ | = | $-1 \cdot 2^{24} \cdot 3^{10} \cdot 5^{6} \cdot 23 \cdot 41 $ |
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| j-invariant: | $j$ | = | \( \frac{1276229915423}{1281494089728} \) | = | $2^{-24} \cdot 3^{-4} \cdot 23^{-1} \cdot 41^{-1} \cdot 10847^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3562856285931420417432205445$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0022605280420370087452182594$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9925476282275606$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.980571312741358$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.5289324684379415842846020197$ |
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| Real period: | $\Omega$ | ≈ | $0.10349401695230415845753847523$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ ( 2^{3} \cdot 3 )\cdot2\cdot2\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.7653615210892692625670527946 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.765361521 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.103494 \cdot 3.528932 \cdot 96}{2^2} \\ & \approx 8.765361521\end{aligned}$$
Modular invariants
Modular form 424350.2.a.co
For more coefficients, see the Downloads section to the right.
| Modular degree: | 10223616 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 4 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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $24$ | $I_{24}$ | split multiplicative | -1 | 1 | 24 | 24 |
| $3$ | $2$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $23$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $41$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 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 | 16.24.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 226320 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \cdot 41 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 160321 & 120720 \\ 179370 & 94591 \end{array}\right),\left(\begin{array}{rr} 376 & 120705 \\ 35775 & 181066 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 120720 \\ 56580 & 56581 \end{array}\right),\left(\begin{array}{rr} 226305 & 16 \\ 226304 & 17 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 226222 & 226307 \end{array}\right),\left(\begin{array}{rr} 75439 & 0 \\ 0 & 226319 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 45263 & 0 \\ 0 & 226319 \end{array}\right),\left(\begin{array}{rr} 68896 & 150885 \\ 216435 & 226306 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 226316 & 226317 \end{array}\right)$.
The torsion field $K:=\Q(E[226320])$ is a degree-$2170850987999232000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/226320\Z)$.
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 | $4$ | \( 212175 = 3^{2} \cdot 5^{2} \cdot 23 \cdot 41 \) |
| $3$ | additive | $8$ | \( 23575 = 5^{2} \cdot 23 \cdot 41 \) |
| $5$ | additive | $14$ | \( 16974 = 2 \cdot 3^{2} \cdot 23 \cdot 41 \) |
| $23$ | nonsplit multiplicative | $24$ | \( 18450 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 41 \) |
| $41$ | nonsplit multiplicative | $42$ | \( 10350 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 23 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 424350co
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 5658g1, its twist by $-15$.
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.