Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-114075x-14857750\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-114075xz^2-14857750z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-114075x-14857750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2695, 138750)$ | $3.1053458113472533016487454719$ | $\infty$ |
Integral points
\((2695,\pm 138750)\)
Invariants
| Conductor: | $N$ | = | \( 478800 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 19$ |
|
| Discriminant: | $\Delta$ | = | $-359100000000000$ | = | $-1 \cdot 2^{11} \cdot 3^{3} \cdot 5^{11} \cdot 7 \cdot 19 $ |
|
| j-invariant: | $j$ | = | \( -\frac{190012163094}{415625} \) | = | $-1 \cdot 2 \cdot 3^{9} \cdot 5^{-5} \cdot 7^{-1} \cdot 13^{6} \cdot 19^{-1}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6763480745389630943134834251$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.038408869358397716134836995415$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0062279724178593$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.559222703039296$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.1053458113472533016487454719$ |
|
| Real period: | $\Omega$ | ≈ | $0.12987184817167807371300107306$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2^{2}\cdot1\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $6.4527519957095510898249103634 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 6.452751996 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.129872 \cdot 3.105346 \cdot 16}{1^2} \\ & \approx 6.452751996\end{aligned}$$
Modular invariants
Modular form 478800.2.a.l
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2887680 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{3}^{*}$ | additive | 1 | 4 | 11 | 0 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $4$ | $I_{5}^{*}$ | additive | 1 | 2 | 11 | 5 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $19$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 15960 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 19 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 9577 & 2 \\ 9577 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 15959 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3991 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15959 & 2 \\ 15958 & 3 \end{array}\right),\left(\begin{array}{rr} 13681 & 2 \\ 13681 & 3 \end{array}\right),\left(\begin{array}{rr} 5321 & 2 \\ 5321 & 3 \end{array}\right),\left(\begin{array}{rr} 4201 & 2 \\ 4201 & 3 \end{array}\right),\left(\begin{array}{rr} 7981 & 2 \\ 7981 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[15960])$ is a degree-$4392005035622400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/15960\Z)$.
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 | $4$ | \( 9975 = 3 \cdot 5^{2} \cdot 7 \cdot 19 \) |
| $3$ | additive | $6$ | \( 53200 = 2^{4} \cdot 5^{2} \cdot 7 \cdot 19 \) |
| $5$ | additive | $14$ | \( 19152 = 2^{4} \cdot 3^{2} \cdot 7 \cdot 19 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 68400 = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 25200 = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 478800l consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 47880d1, 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.