Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-x^2+258x-3834\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-x^2z+258xz^2-3834z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+4125x-241250\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(33, 183)$ | $1.9432366744573975877651280536$ | $\infty$ |
| $(19, 78)$ | $0$ | $3$ |
Integral points
\( \left(19, 78\right) \), \( \left(19, -97\right) \), \( \left(33, 183\right) \), \( \left(33, -216\right) \), \( \left(369, 6903\right) \), \( \left(369, -7272\right) \)
Invariants
| Conductor: | $N$ | = | \( 3150 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7$ |
|
| Discriminant: | $\Delta$ | = | $-7235156250$ | = | $-1 \cdot 2 \cdot 3^{3} \cdot 5^{8} \cdot 7^{3} $ |
|
| j-invariant: | $j$ | = | \( \frac{179685}{686} \) | = | $2^{-1} \cdot 3^{3} \cdot 5 \cdot 7^{-3} \cdot 11^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.57463927845886548727749399882$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.77297240199756218530515686590$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.8976142640226761$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.7233979032854823$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.9432366744573975877651280536$ |
|
| Real period: | $\Omega$ | ≈ | $0.67231428673888639633559472379$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 18 $ = $ 1\cdot2\cdot3\cdot3 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $2.6129315575053416805068522745 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 2.612931558 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.672314 \cdot 1.943237 \cdot 18}{3^2} \\ & \approx 2.612931558\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1440 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 |
|---|---|---|
| $3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 168 = 2^{3} \cdot 3 \cdot 7 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 165 & 166 \\ 158 & 161 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 163 & 6 \\ 162 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 78 & 97 \\ 115 & 54 \end{array}\right),\left(\begin{array}{rr} 127 & 6 \\ 45 & 19 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 85 & 6 \\ 87 & 19 \end{array}\right),\left(\begin{array}{rr} 73 & 6 \\ 51 & 19 \end{array}\right)$.
The torsion field $K:=\Q(E[168])$ is a degree-$9289728$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/168\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| $3$ | additive | $6$ | \( 50 = 2 \cdot 5^{2} \) |
| $5$ | additive | $14$ | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| $7$ | split multiplicative | $8$ | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 3150.o
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 3150.h2, its twist by $-15$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.4200.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.2963520000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.21870000.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.3.422033639910750000.14 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.20162971139764174848000000000000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | 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 | nonsplit | add | add | split | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | - | - | 2 | 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 | 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.