Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3+25x-375\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3+25xz^2-375z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+32373x-17593146\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(40, 235)$ | $0$ | $6$ |
Integral points
\( \left(6, -3\right) \), \( \left(10, 25\right) \), \( \left(10, -35\right) \), \( \left(40, 235\right) \), \( \left(40, -275\right) \)
Invariants
| Conductor: | $N$ | = | \( 510 \) | = | $2 \cdot 3 \cdot 5 \cdot 17$ |
|
| Discriminant: | $\Delta$ | = | $-62424000$ | = | $-1 \cdot 2^{6} \cdot 3^{3} \cdot 5^{3} \cdot 17^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{1723683599}{62424000} \) | = | $2^{-6} \cdot 3^{-3} \cdot 5^{-3} \cdot 11^{3} \cdot 17^{-2} \cdot 109^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.17569996747436741068407480315$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.17569996747436741068407480315$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9764195258143714$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.072248834492683$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.94662416098739111083987584867$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 108 $ = $ ( 2 \cdot 3 )\cdot3\cdot3\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $6$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.8398724829621733325196275460 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 2.839872483 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.946624 \cdot 1.000000 \cdot 108}{6^2} \\ & \approx 2.839872483\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 144 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $17$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 | 2.3.0.1 |
| $3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 1990 & 2031 \end{array}\right),\left(\begin{array}{rr} 1706 & 11 \\ 669 & 2020 \end{array}\right),\left(\begin{array}{rr} 10 & 3 \\ 381 & 2032 \end{array}\right),\left(\begin{array}{rr} 2029 & 12 \\ 2028 & 13 \end{array}\right),\left(\begin{array}{rr} 241 & 12 \\ 1446 & 73 \end{array}\right),\left(\begin{array}{rr} 435 & 88 \\ 398 & 77 \end{array}\right),\left(\begin{array}{rr} 1021 & 12 \\ 6 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[2040])$ is a degree-$28877783040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2040\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 15 = 3 \cdot 5 \) |
| $3$ | split multiplicative | $4$ | \( 17 \) |
| $5$ | split multiplicative | $6$ | \( 102 = 2 \cdot 3 \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 510.g
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | 4.2.277440.1 | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.2255067.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.3370896000000.39 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.17318914560000.5 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $9$ | 9.3.1198435061547000000.11 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.0.4847332264014920663362080375000000000000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | 17 |
|---|---|---|---|---|
| Reduction type | split | split | split | nonsplit |
| $\lambda$-invariant(s) | 4 | 3 | 1 | 0 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.