Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+75x+750\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+75xz^2+750z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+75x+750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(10, 50)$ | $0.95250316419196536870233013993$ | $\infty$ |
Integral points
\((10,\pm 50)\)
Invariants
| Conductor: | $N$ | = | \( 86400 \) | = | $2^{7} \cdot 3^{3} \cdot 5^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-270000000$ | = | $-1 \cdot 2^{7} \cdot 3^{3} \cdot 5^{7} $ |
|
| j-invariant: | $j$ | = | \( \frac{864}{5} \) | = | $2^{5} \cdot 3^{3} \cdot 5^{-1}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.29979202198785399909532114647$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1839158617228583748805885669$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.6769924925288455$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.3545260112867767$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.95250316419196536870233013993$ |
|
| Real period: | $\Omega$ | ≈ | $1.2587755551031696539633215686$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot1\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.7959507969730670215108367454 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.795950797 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.258776 \cdot 0.952503 \cdot 4}{1^2} \\ & \approx 4.795950797\end{aligned}$$
Modular invariants
Modular form 86400.2.a.g
For more coefficients, see the Downloads section to the right.
| Modular degree: | 23040 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 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 | 7 | 7 | 0 |
| $3$ | $1$ | $II$ | additive | 1 | 3 | 3 | 0 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 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 \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 31 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 119 & 2 \\ 118 & 3 \end{array}\right),\left(\begin{array}{rr} 61 & 2 \\ 61 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 119 & 0 \end{array}\right),\left(\begin{array}{rr} 97 & 2 \\ 97 & 3 \end{array}\right),\left(\begin{array}{rr} 41 & 2 \\ 41 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$17694720$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
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 | $4$ | \( 675 = 3^{3} \cdot 5^{2} \) |
| $3$ | additive | $6$ | \( 3200 = 2^{7} \cdot 5^{2} \) |
| $5$ | additive | $18$ | \( 3456 = 2^{7} \cdot 3^{3} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 86400n consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 17280u1, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.1080.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.139968000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.143327232000000.19 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.