Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-2674408x+1525939312\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-2674408xz^2+1525939312z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-216627075x+1111759877250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-518, 52650)$ | $3.6456539486809605473842994863$ | $\infty$ |
| $(697, 0)$ | $0$ | $2$ |
| $(1172, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1868, 0\right) \), \((-518,\pm 52650)\), \( \left(697, 0\right) \), \( \left(1172, 0\right) \), \((3908,\pm 225264)\)
Invariants
| Conductor: | $N$ | = | \( 433200 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $219497262393600000000$ | = | $2^{14} \cdot 3^{6} \cdot 5^{8} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{702595369}{72900} \) | = | $2^{-2} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{3} \cdot 127^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6383718769625399066051291464$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.33171374939767582011699635762$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0045688963827404$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.315549295871445$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.6456539486809605473842994863$ |
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| Real period: | $\Omega$ | ≈ | $0.17195131756456457096260153225$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.0149999988811890909036146998 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.014999999 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.171951 \cdot 3.645654 \cdot 128}{4^2} \\ & \approx 5.014999999\end{aligned}$$
Modular invariants
Modular form 433200.2.a.n
For more coefficients, see the Downloads section to the right.
| Modular degree: | 15925248 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{6}^{*}$ | additive | -1 | 4 | 14 | 2 |
| $3$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $19$ | $4$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
| $3$ | 3B | 3.4.0.1 |
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 $384$, genus $5$, and generators
$\left(\begin{array}{rr} 9 & 4 \\ 2264 & 2273 \end{array}\right),\left(\begin{array}{rr} 1823 & 114 \\ 0 & 2279 \end{array}\right),\left(\begin{array}{rr} 1521 & 1444 \\ 1102 & 609 \end{array}\right),\left(\begin{array}{rr} 2269 & 12 \\ 2268 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1027 & 2166 \\ 114 & 115 \end{array}\right),\left(\begin{array}{rr} 569 & 228 \\ 1254 & 1367 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 359 & 0 \\ 0 & 2279 \end{array}\right)$.
The torsion field $K:=\Q(E[2280])$ is a degree-$11346739200$ 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 | $2$ | \( 9025 = 5^{2} \cdot 19^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 144400 = 2^{4} \cdot 5^{2} \cdot 19^{2} \) |
| $5$ | additive | $18$ | \( 17328 = 2^{4} \cdot 3 \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 433200n
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 30a2, its twist by $380$.
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.