Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-4537168x+1922830868\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-4537168xz^2+1922830868z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-367510635x+1402846234650\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-772, 70470)$ | $2.8551797092832867776038036056$ | $\infty$ |
| $(443, 0)$ | $0$ | $2$ |
Integral points
\((-772,\pm 70470)\), \( \left(443, 0\right) \), \((3692,\pm 188442)\)
Invariants
| Conductor: | $N$ | = | \( 433200 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $4377969477235805184000$ | = | $2^{13} \cdot 3^{14} \cdot 5^{3} \cdot 19^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{428831641421}{181752822} \) | = | $2^{-1} \cdot 3^{-14} \cdot 19^{-1} \cdot 7541^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8493017230236395590088859846$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.28157557477194892593695031389$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9846422224017274$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.437725913011962$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.8551797092832867776038036056$ |
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| Real period: | $\Omega$ | ≈ | $0.12473441097011396591055695942$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 112 $ = $ 2\cdot( 2 \cdot 7 )\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $9.9718964590356162863104175469 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.971896459 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.124734 \cdot 2.855180 \cdot 112}{2^2} \\ & \approx 9.971896459\end{aligned}$$
Modular invariants
Modular form 433200.2.a.il
For more coefficients, see the Downloads section to the right.
| Modular degree: | 23224320 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{5}^{*}$ | additive | -1 | 4 | 13 | 1 |
| $3$ | $14$ | $I_{14}$ | split multiplicative | -1 | 1 | 14 | 14 |
| $5$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $19$ | $2$ | $I_{1}^{*}$ | 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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 761 & 4 \\ 1522 & 9 \end{array}\right),\left(\begin{array}{rr} 1562 & 1 \\ 359 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 1139 & 0 \end{array}\right),\left(\begin{array}{rr} 2277 & 4 \\ 2276 & 5 \end{array}\right),\left(\begin{array}{rr} 1828 & 1 \\ 1823 & 0 \end{array}\right),\left(\begin{array}{rr} 1996 & 289 \\ 1425 & 856 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[2280])$ is a degree-$363095654400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2280\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 1805 = 5 \cdot 19^{2} \) |
| $3$ | split multiplicative | $4$ | \( 144400 = 2^{4} \cdot 5^{2} \cdot 19^{2} \) |
| $5$ | additive | $10$ | \( 17328 = 2^{4} \cdot 3 \cdot 19^{2} \) |
| $7$ | good | $2$ | \( 144400 = 2^{4} \cdot 5^{2} \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 433200il
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 2850n2, its twist by $76$.
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.