Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+x^2-1112x+13908\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+x^2z-1112xz^2+13908z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-90099x+10409202\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(22, 24)$ | $0$ | $4$ |
Integral points
\( \left(19, 0\right) \), \((22,\pm 24)\)
Invariants
| Conductor: | $N$ | = | \( 624 \) | = | $2^{4} \cdot 3 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $4313088$ | = | $2^{12} \cdot 3^{4} \cdot 13 $ |
|
| j-invariant: | $j$ | = | \( \frac{37159393753}{1053} \) | = | $3^{-4} \cdot 13^{-1} \cdot 47^{3} \cdot 71^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.37478071843155785646845510429$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.31836646212838745294877701717$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.1161605159157866$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.073879063401496$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $2.2865886336506679916867291861$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2^{2}\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.2865886336506679916867291861 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 2.286588634 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.286589 \cdot 1.000000 \cdot 16}{4^2} \\ & \approx 2.286588634\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 256 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | -1 | 4 | 12 | 0 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $13$ | $1$ | $I_{1}$ | split 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 |
|---|---|---|
| $2$ | 2B | 4.12.0.7 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 312 = 2^{3} \cdot 3 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 306 & 307 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 269 & 272 \\ 94 & 267 \end{array}\right),\left(\begin{array}{rr} 43 & 42 \\ 130 & 283 \end{array}\right),\left(\begin{array}{rr} 305 & 8 \\ 304 & 9 \end{array}\right),\left(\begin{array}{rr} 268 & 1 \\ 215 & 6 \end{array}\right),\left(\begin{array}{rr} 209 & 8 \\ 212 & 33 \end{array}\right)$.
The torsion field $K:=\Q(E[312])$ is a degree-$40255488$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/312\Z)$.
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 | $2$ | \( 13 \) |
| $3$ | split multiplicative | $4$ | \( 208 = 2^{4} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 48 = 2^{4} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 624.i
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 39.a1, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.4.29952.1 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.19770609664.3 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1601419382784.41 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.8.151613669376.2 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.1295230839552.1 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 13 |
|---|---|---|---|
| Reduction type | add | split | split |
| $\lambda$-invariant(s) | - | 1 | 1 |
| $\mu$-invariant(s) | - | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.