Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-289501x-22961352\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-289501xz^2-22961352z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-375192675x-1070159249250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-493, 246)$ | $0$ | $2$ |
Integral points
\( \left(-493, 246\right) \)
Invariants
| Conductor: | $N$ | = | \( 8450 \) | = | $2 \cdot 5^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $1325562421625000000$ | = | $2^{6} \cdot 5^{9} \cdot 13^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{16194277}{8000} \) | = | $2^{-6} \cdot 5^{-3} \cdot 11^{3} \cdot 23^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1704642584644924884879057813$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.55796671584871025085258946648$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9455380306483595$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.45695281202063$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.21650421619166013511529451009$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.43300843238332027023058902018 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.433008432 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.216504 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 0.433008432\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 134784 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $13$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3Ns | 3.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $144$, genus $5$, and generators
$\left(\begin{array}{rr} 10 & 3 \\ 1221 & 1552 \end{array}\right),\left(\begin{array}{rr} 781 & 12 \\ 0 & 1301 \end{array}\right),\left(\begin{array}{rr} 7 & 12 \\ 144 & 247 \end{array}\right),\left(\begin{array}{rr} 492 & 5 \\ 187 & 1488 \end{array}\right),\left(\begin{array}{rr} 781 & 12 \\ 6 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 5 & 12 \\ 1512 & 1445 \end{array}\right),\left(\begin{array}{rr} 725 & 588 \\ 1338 & 707 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1549 & 12 \\ 1548 & 13 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$6440878080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $4$ | \( 325 = 5^{2} \cdot 13 \) |
| $3$ | good | $2$ | \( 4225 = 5^{2} \cdot 13^{2} \) |
| $5$ | additive | $18$ | \( 338 = 2 \cdot 13^{2} \) |
| $13$ | additive | $62$ | \( 50 = 2 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 8450.b
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1690.d2, its twist by $65$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{65}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.703040.4 | \(\Z/4\Z\) | not in database |
| $4$ | 4.2.164775.1 | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.494325.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.4.77228944000000.9 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.12356631040000.10 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.244357205625.4 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.4.27150800625.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.244357205625.7 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | 5 | 13 |
|---|---|---|---|
| Reduction type | nonsplit | add | add |
| $\lambda$-invariant(s) | 2 | - | - |
| $\mu$-invariant(s) | 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 $0$.