Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-10183x-392138\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-10183xz^2-392138z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-824850x-288343125\)
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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 |
|---|---|---|
| $(1078/9, 8200/27)$ | $5.7679187190959065703648844678$ | $\infty$ |
| $(726, 19348)$ | $6.6643178224227327012968786705$ | $\infty$ |
| $(-58, 0)$ | $0$ | $2$ |
Integral points
\( \left(-58, 0\right) \), \((726,\pm 19348)\)
Invariants
| Conductor: | $N$ | = | \( 488400 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $305250000$ | = | $2^{4} \cdot 3 \cdot 5^{6} \cdot 11 \cdot 37 $ |
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| j-invariant: | $j$ | = | \( \frac{467147020288}{1221} \) | = | $2^{11} \cdot 3^{-1} \cdot 11^{-1} \cdot 13^{3} \cdot 37^{-1} \cdot 47^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.86409582456387492286398446839$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.17167219183982370090880590538$ |
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| $abc$ quality: | $Q$ | ≈ | $0.888149319585553$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.000187496563337$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $30.336195315464319638463705250$ |
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| Real period: | $\Omega$ | ≈ | $0.47525547952035326044640039836$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot1\cdot2^{2}\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $14.417443051474089479127851874 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.417443051 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.475255 \cdot 30.336195 \cdot 4}{2^2} \\ & \approx 14.417443051\end{aligned}$$
Modular invariants
Modular form 488400.2.a.cb
For more coefficients, see the Downloads section to the right.
| Modular degree: | 458752 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | 1 | 4 | 4 | 0 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $37$ | $1$ | $I_{1}$ | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 48840 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 37 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 48834 & 48835 \end{array}\right),\left(\begin{array}{rr} 29303 & 0 \\ 0 & 48839 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 26056 & 39075 \\ 16285 & 29306 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 32971 & 32970 \\ 20770 & 25651 \end{array}\right),\left(\begin{array}{rr} 28079 & 28080 \\ 3650 & 47609 \end{array}\right),\left(\begin{array}{rr} 6076 & 29305 \\ 32495 & 6 \end{array}\right),\left(\begin{array}{rr} 48833 & 8 \\ 48832 & 9 \end{array}\right),\left(\begin{array}{rr} 896 & 39075 \\ 31085 & 29306 \end{array}\right)$.
The torsion field $K:=\Q(E[48840])$ is a degree-$17733591760896000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/48840\Z)$.
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 | $2$ | \( 30525 = 3 \cdot 5^{2} \cdot 11 \cdot 37 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 162800 = 2^{4} \cdot 5^{2} \cdot 11 \cdot 37 \) |
| $5$ | additive | $14$ | \( 19536 = 2^{4} \cdot 3 \cdot 11 \cdot 37 \) |
| $11$ | split multiplicative | $12$ | \( 44400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 37 \) |
| $37$ | nonsplit multiplicative | $38$ | \( 13200 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 488400.cb
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 9768.f3, its twist by $-20$.
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.