Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-965900x-140310000\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-965900xz^2-140310000z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-965900x-140310000\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-900, 0)$ | $0$ | $2$ |
Integral points
\( \left(-900, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 123200 \) | = | $2^{6} \cdot 5^{2} \cdot 7 \cdot 11$ |
|
| Discriminant: | $\Delta$ | = | $49168782688256000000$ | = | $2^{21} \cdot 5^{6} \cdot 7 \cdot 11^{8} $ |
|
| j-invariant: | $j$ | = | \( \frac{24331017010833}{12004097336} \) | = | $2^{-3} \cdot 3^{3} \cdot 7^{-1} \cdot 11^{-8} \cdot 13^{3} \cdot 743^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4716128052435977075364511398$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.62717307818662955611022329100$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.1340815776703688$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.517831264323862$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.16017134227647134182461241655$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2^{2}\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $1.2813707382117707345968993324 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 1.281370738 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.160171 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 1.281370738\end{aligned}$$
Modular invariants
Modular form 123200.2.a.dc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2359296 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{11}^{*}$ | additive | -1 | 6 | 21 | 3 |
| $5$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $11$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
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 | 8.12.0.9 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 280 = 2^{3} \cdot 5 \cdot 7 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 273 & 8 \\ 272 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 104 & 125 \\ 255 & 114 \end{array}\right),\left(\begin{array}{rr} 144 & 55 \\ 175 & 184 \end{array}\right),\left(\begin{array}{rr} 223 & 0 \\ 0 & 279 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 274 & 275 \end{array}\right),\left(\begin{array}{rr} 236 & 225 \\ 55 & 6 \end{array}\right)$.
The torsion field $K:=\Q(E[280])$ is a degree-$30965760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/280\Z)$.
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 | $4$ | \( 175 = 5^{2} \cdot 7 \) |
| $5$ | additive | $14$ | \( 4928 = 2^{6} \cdot 7 \cdot 11 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 17600 = 2^{6} \cdot 5^{2} \cdot 11 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 11200 = 2^{6} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 123200dx
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 154b4, its twist by $-40$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{14}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-70}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{14})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.308409794560000.49 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1204725760000.11 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.513802240000.15 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 5 | 7 | 11 |
|---|---|---|---|---|
| Reduction type | add | add | nonsplit | nonsplit |
| $\lambda$-invariant(s) | - | - | 0 | 0 |
| $\mu$-invariant(s) | - | - | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.