Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-51x-40\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-51xz^2-40z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-4158x-41607\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-1, 3\right) \) | $1.5932486290473960404252771416$ | $\infty$ |
| \( \left(8, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-1:3:1]\) | $1.5932486290473960404252771416$ | $\infty$ |
| \([8:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-12, 81\right) \) | $1.5932486290473960404252771416$ | $\infty$ |
| \( \left(69, 0\right) \) | $0$ | $2$ |
Integral points
\((-1,\pm 3)\), \( \left(8, 0\right) \)
\([-1:\pm 3:1]\), \([8:0:1]\)
\((-1,\pm 3)\), \( \left(8, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 1160 \) | = | $2^{3} \cdot 5 \cdot 29$ |
|
| Minimal Discriminant: | $\Delta$ | = | $7250000$ | = | $2^{4} \cdot 5^{6} \cdot 29 $ |
|
| j-invariant: | $j$ | = | \( \frac{934979584}{453125} \) | = | $2^{11} \cdot 5^{-6} \cdot 7^{3} \cdot 11^{3} \cdot 29^{-1}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.0099235765880868730901238202607$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.22112548359856156338228688689$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.894034150371482$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.320300733848215$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.5932486290473960404252771416$ |
|
| Real period: | $\Omega$ | ≈ | $1.8724484013366199453989947129$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $2.9832758483915581356249991807 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 2.983275848 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.872448 \cdot 1.593249 \cdot 4}{2^2} \\ & \approx 2.983275848\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 192 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | 1 | 3 | 4 | 0 |
| $5$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $29$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 580 = 2^{2} \cdot 5 \cdot 29 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 577 & 4 \\ 576 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 437 & 146 \\ 144 & 435 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 62 & 1 \\ 259 & 0 \end{array}\right),\left(\begin{array}{rr} 117 & 4 \\ 234 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[580])$ is a degree-$2619187200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/580\Z)$.
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 | $2$ | \( 29 \) |
| $3$ | good | $2$ | \( 232 = 2^{3} \cdot 29 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 232 = 2^{3} \cdot 29 \) |
| $29$ | nonsplit multiplicative | $30$ | \( 40 = 2^{3} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 1160.c
consists of 2 curves linked by isogenies of
degree 2.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{29}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.4.725.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.8.442050625.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.60909908070400.12 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.1583947312128.7 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | add | ord | nonsplit | ord | ss | ord | ord | ss | ss | nonsplit | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | 1 | 1 | 1,1 | 3 | 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 | 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.