Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-33146892x-74051194896\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-33146892xz^2-74051194896z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-33146892x-74051194896\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(20151035905599111846796356225/46425817573773812164, 2860527091529411170055708920120383478056801/316329242106169052888451190712)$ | $61.509499229964313543549145534$ | $\infty$ |
| $(6654, 0)$ | $0$ | $2$ |
Integral points
\( \left(6654, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 411840 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $-38088058244014622638080$ | = | $-1 \cdot 2^{38} \cdot 3^{6} \cdot 5 \cdot 11^{3} \cdot 13^{4} $ |
|
| j-invariant: | $j$ | = | \( -\frac{21075830718885163521}{199306463150080} \) | = | $-1 \cdot 2^{-20} \cdot 3^{3} \cdot 5^{-1} \cdot 11^{-3} \cdot 13^{-4} \cdot 181^{3} \cdot 5087^{3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1540828105331219750040045376$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.5650558953591491651805337370$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0038593376627314$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.917796381847909$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $61.509499229964313543549145534$ |
|
| Real period: | $\Omega$ | ≈ | $0.031442270506839908267956249588$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2\cdot1\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $7.7359932541151959364958689581 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 7.735993254 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.031442 \cdot 61.509499 \cdot 16}{2^2} \\ & \approx 7.735993254\end{aligned}$$
Modular invariants
Modular form 411840.2.a.jo
For more coefficients, see the Downloads section to the right.
| Modular degree: | 41287680 |
|
| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{28}^{*}$ | additive | 1 | 6 | 38 | 20 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $13$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 \( 17160 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 4997 & 10722 \\ 12150 & 9293 \end{array}\right),\left(\begin{array}{rr} 17153 & 8 \\ 17152 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2641 & 11448 \\ 16284 & 11473 \end{array}\right),\left(\begin{array}{rr} 6868 & 5721 \\ 8031 & 11446 \end{array}\right),\left(\begin{array}{rr} 14044 & 5721 \\ 2103 & 11446 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 17154 & 17155 \end{array}\right),\left(\begin{array}{rr} 7873 & 2148 \\ 13590 & 16447 \end{array}\right),\left(\begin{array}{rr} 5719 & 0 \\ 0 & 17159 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[17160])$ is a degree-$255058771968000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/17160\Z)$.
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 | $2$ | \( 495 = 3^{2} \cdot 5 \cdot 11 \) |
| $3$ | additive | $6$ | \( 4160 = 2^{6} \cdot 5 \cdot 13 \) |
| $5$ | split multiplicative | $6$ | \( 82368 = 2^{6} \cdot 3^{2} \cdot 11 \cdot 13 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 37440 = 2^{6} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 31680 = 2^{6} \cdot 3^{2} \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 411840jo
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1430h1, its twist by $-24$.
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.