Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-6591x+153790\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-6591xz^2+153790z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-6591x+153790\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(17, 216)$ | $1.9606576470820368081528882122$ | $\infty$ |
| $(101, 720)$ | $2.9700369171280745358731764119$ | $\infty$ |
| $(26, 0)$ | $0$ | $2$ |
| $(65, 0)$ | $0$ | $2$ |
Integral points
\( \left(-91, 0\right) \), \((-82,\pm 378)\), \((17,\pm 216)\), \( \left(26, 0\right) \), \( \left(65, 0\right) \), \((78,\pm 338)\), \((101,\pm 720)\), \((234,\pm 3380)\), \((533,\pm 12168)\), \((1937,\pm 85176)\)
Invariants
| Conductor: | $N$ | = | \( 24336 \) | = | $2^{4} \cdot 3^{2} \cdot 13^{2}$ |
|
| Discriminant: | $\Delta$ | = | $8107185625344$ | = | $2^{8} \cdot 3^{8} \cdot 13^{6} $ |
|
| j-invariant: | $j$ | = | \( \frac{35152}{9} \) | = | $2^{4} \cdot 3^{-2} \cdot 13^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1864285369849659410653556624$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1074504064531541456038320911$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.972547111469975$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.761887869806381$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
|
| Mordell-Weil rank: | $r$ | = | $ 2$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.0946584227856233274083264590$ |
|
| Real period: | $\Omega$ | ≈ | $0.69063773857179988081401932767$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $7.0371267438168712653965581926 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 7.037126744 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.690638 \cdot 5.094658 \cdot 32}{4^2} \\ & \approx 7.037126744\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 36864 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | 1 | 4 | 8 | 0 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $13$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 |
|---|---|---|
| $2$ | 2Cs | 8.48.0.138 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 312 = 2^{3} \cdot 3 \cdot 13 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 25 & 286 \\ 234 & 77 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 209 & 0 \\ 208 & 53 \end{array}\right),\left(\begin{array}{rr} 167 & 0 \\ 0 & 311 \end{array}\right),\left(\begin{array}{rr} 259 & 208 \\ 26 & 285 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 308 & 309 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 305 & 8 \\ 304 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[312])$ is a degree-$10063872$ 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 | $2$ | \( 1521 = 3^{2} \cdot 13^{2} \) |
| $3$ | additive | $8$ | \( 2704 = 2^{4} \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 144 = 2^{4} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 24336.o
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 24.a4, its twist by $156$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{39}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{-13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.592240896.1 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.85282689024.33 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.151613669376.8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.8.151613669376.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.20723693432832.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.22986704741655040229376.1 | \(\Z/4\Z \oplus \Z/8\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | ord | ss | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | - | 2 | 2,2 | 2 | - | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2,2 |
| $\mu$-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.