Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-70245x+7189389\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-70245xz^2+7189389z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-5689872x+5223994992\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(156, 57)$ | $0.22628491623358108560693338282$ | $\infty$ |
Integral points
\((-300,\pm 1083)\), \((156,\pm 57)\)
Invariants
| Conductor: | $N$ | = | \( 912 \) | = | $2^{4} \cdot 3 \cdot 19$ |
|
| Discriminant: | $\Delta$ | = | $-91278913536$ | = | $-1 \cdot 2^{12} \cdot 3^{2} \cdot 19^{5} $ |
|
| j-invariant: | $j$ | = | \( -\frac{9358714467168256}{22284891} \) | = | $-1 \cdot 2^{12} \cdot 3^{-2} \cdot 19^{-5} \cdot 13171^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3445544317883737969809755518$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.65140725122842848756374343034$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0683303788164564$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.6160851629187105$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.22628491623358108560693338282$ |
|
| Real period: | $\Omega$ | ≈ | $0.92672254267264968630520894650$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 10 $ = $ 1\cdot2\cdot5 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $2.0970333294045180733689571813 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 2.097033329 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.926723 \cdot 0.226285 \cdot 10}{1^2} \\ & \approx 2.097033329\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2400 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| 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 | 4 | 12 | 0 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $19$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
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 |
|---|---|---|
| $5$ | 5B.4.2 | 5.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 380 = 2^{2} \cdot 5 \cdot 19 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 189 & 0 \\ 0 & 379 \end{array}\right),\left(\begin{array}{rr} 7 & 10 \\ 150 & 171 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 199 & 370 \\ 200 & 369 \end{array}\right),\left(\begin{array}{rr} 254 & 375 \\ 295 & 194 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 325 & 261 \end{array}\right),\left(\begin{array}{rr} 371 & 10 \\ 370 & 11 \end{array}\right),\left(\begin{array}{rr} 374 & 367 \\ 245 & 309 \end{array}\right)$.
The torsion field $K:=\Q(E[380])$ is a degree-$118195200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/380\Z)$.
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 | $2$ | \( 19 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 304 = 2^{4} \cdot 19 \) |
| $5$ | good | $2$ | \( 48 = 2^{4} \cdot 3 \) |
| $19$ | split multiplicative | $20$ | \( 48 = 2^{4} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 912f
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 57c2, its twist by $-4$.
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 |
|---|---|---|---|
| $3$ | 3.1.76.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\zeta_{20})^+\) | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.109744.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.5910009391872.5 | \(\Z/3\Z\) | not in database |
| $10$ | 10.0.65610000000000.7 | \(\Z/5\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | 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 | nonsplit | ord | ord | ord | ord | ord | split | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | 3 | 1 | 1 | 1 | 1 | 2 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.