Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-234x+1352\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-234xz^2+1352z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-302643x+63998478\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(8, -1)$ | $1.5187021867268516013726836430$ | $\infty$ |
| $(23, 79)$ | $0$ | $6$ |
Integral points
\( \left(-16, 40\right) \), \( \left(-16, -25\right) \), \( \left(-5, 51\right) \), \( \left(-5, -47\right) \), \( \left(8, -1\right) \), \( \left(8, -8\right) \), \( \left(9, -5\right) \), \( \left(10, 1\right) \), \( \left(10, -12\right) \), \( \left(23, 79\right) \), \( \left(23, -103\right) \)
Invariants
| Conductor: | $N$ | = | \( 910 \) | = | $2 \cdot 5 \cdot 7 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $2153060$ | = | $2^{2} \cdot 5 \cdot 7^{2} \cdot 13^{3} $ |
|
| j-invariant: | $j$ | = | \( \frac{1408317602329}{2153060} \) | = | $2^{-2} \cdot 5^{-1} \cdot 7^{-2} \cdot 11^{3} \cdot 13^{-3} \cdot 1019^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.11483821143717816948776519659$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.11483821143717816948776519659$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.895954553441358$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.1056203671299185$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.5187021867268516013726836430$ |
|
| Real period: | $\Omega$ | ≈ | $2.6028837105666664734077006574$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2\cdot1\cdot2\cdot3 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $6$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $1.3176683943444326217031467282 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 1.317668394 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.602884 \cdot 1.518702 \cdot 12}{6^2} \\ & \approx 1.317668394\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 288 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $3$ | $I_{3}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
| $3$ | 3B.1.1 | 3.8.0.1 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 7801 & 12 \\ 3126 & 73 \end{array}\right),\left(\begin{array}{rr} 5919 & 7738 \\ 3206 & 5021 \end{array}\right),\left(\begin{array}{rr} 10 & 3 \\ 4341 & 10912 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 10870 & 10911 \end{array}\right),\left(\begin{array}{rr} 5461 & 12 \\ 6 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 7289 & 2 \\ 1878 & 13 \end{array}\right),\left(\begin{array}{rr} 10909 & 12 \\ 10908 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 9250 & 3 \\ 6693 & 10912 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$19477215313920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 | $4$ | \( 65 = 5 \cdot 13 \) |
| $3$ | good | $2$ | \( 70 = 2 \cdot 5 \cdot 7 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 182 = 2 \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 130 = 2 \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 70 = 2 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 910c
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{65}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | 4.4.203840.1 | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.648270000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.8.175551900160000.1 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $9$ | 9.3.2281096017943470000.2 | \(\Z/18\Z\) | 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 |
| $18$ | 18.6.35724586555129342722603843947054062500000000.2 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | ord | nonsplit | split | ss | split | ss | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 4 | 1 | 1 | 2 | 1,1 | 6 | 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 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.
Additional information
This elliptic curve is a Weierstrass model for the curve $$X/(Y+Z) + Y/(X+Z) + Z/(X+Y) = 4.$$ The associated diophantine equation was popularized by a problem that appears in a well-known 'fruits meme'.