Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-18963x+388962\)
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(homogenize, simplify) |
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\(y^2z=x^3-18963xz^2+388962z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-18963x+388962\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-21, 882)$ | $1.3066351469328317521923486032$ | $\infty$ |
| $(9, 468)$ | $1.5165922095787116587939927302$ | $\infty$ |
| $(21, 0)$ | $0$ | $2$ |
| $(126, 0)$ | $0$ | $2$ |
Integral points
\( \left(-147, 0\right) \), \((-126,\pm 882)\), \((-119,\pm 980)\), \((-69,\pm 1170)\), \((-21,\pm 882)\), \((9,\pm 468)\), \( \left(21, 0\right) \), \( \left(126, 0\right) \), \((141,\pm 720)\), \((142,\pm 748)\), \((147,\pm 882)\), \((217,\pm 2548)\), \((382,\pm 6992)\), \((441,\pm 8820)\), \((501,\pm 10800)\), \((1491,\pm 57330)\), \((2037,\pm 91728)\), \((71001,\pm 18918900)\), \((105987,\pm 34504722)\), \((2154741,\pm 3162952080)\)
Invariants
| Conductor: | $N$ | = | \( 458640 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $371058545894400$ | = | $2^{10} \cdot 3^{6} \cdot 5^{2} \cdot 7^{6} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{8586756}{4225} \) | = | $2^{2} \cdot 3^{3} \cdot 5^{-2} \cdot 13^{-2} \cdot 43^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4888083042243473676182806101$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.61107556510398522181304514797$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9666197140470211$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.157737965003693$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.9072715010665552166874474046$ |
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| Real period: | $\Omega$ | ≈ | $0.47608307521928839183300991011$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 256 $ = $ 2^{2}\cdot2^{2}\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $14.528314904093982199610672213 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.528314904 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.476083 \cdot 1.907272 \cdot 256}{4^2} \\ & \approx 14.528314904\end{aligned}$$
Modular invariants
Modular form 458640.2.a.x
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1572864 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $3$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 2729 & 8316 \\ 9618 & 5711 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7279 & 0 \\ 0 & 10919 \end{array}\right),\left(\begin{array}{rr} 8737 & 2604 \\ 2394 & 5209 \end{array}\right),\left(\begin{array}{rr} 3119 & 0 \\ 0 & 10919 \end{array}\right),\left(\begin{array}{rr} 1303 & 6762 \\ 4158 & 4159 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 4201 & 6762 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10917 & 4 \\ 10916 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$38954430627840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 | $2$ | \( 441 = 3^{2} \cdot 7^{2} \) |
| $3$ | additive | $6$ | \( 50960 = 2^{4} \cdot 5 \cdot 7^{2} \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 91728 = 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 35280 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 458640x
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 520a2, its twist by $-84$.
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.