Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-6007x-181724\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-6007xz^2-181724z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7785747x-8361732114\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{181}{4}, \frac{181}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-362:181:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-1614, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 285 \) | = | $3 \cdot 5 \cdot 19$ |
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| Minimal Discriminant: | $\Delta$ | = | $146611125$ | = | $3^{2} \cdot 5^{3} \cdot 19^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{23977812996389881}{146611125} \) | = | $3^{-2} \cdot 5^{-3} \cdot 19^{-4} \cdot 288361^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.75413885482685399943231616098$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.75413885482685399943231616098$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0982581996716574$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.67244184577228$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
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: | $\Omega$ | ≈ | $0.54228287843457483324645988650$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2\cdot3\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.6268486353037244997393796595 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.626848635 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.542283 \cdot 1.000000 \cdot 12}{2^2} \\ & \approx 1.626848635\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 288 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $19$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 | 4.12.0.8 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 760 = 2^{3} \cdot 5 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 99 & 98 \\ 298 & 675 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 753 & 8 \\ 752 & 9 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 754 & 755 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 616 & 3 \\ 5 & 2 \end{array}\right),\left(\begin{array}{rr} 401 & 8 \\ 84 & 33 \end{array}\right),\left(\begin{array}{rr} 483 & 476 \\ 522 & 101 \end{array}\right)$.
The torsion field $K:=\Q(E[760])$ is a degree-$1891123200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/760\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 5 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 19 \) |
| $5$ | split multiplicative | $6$ | \( 57 = 3 \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 15 = 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 285c
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | 2.0.4.1-81225.2-c4 |
| $4$ | \(\Q(i, \sqrt{5})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt[4]{45})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.5184000000.15 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.133448704000000.11 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.17294952038400.78 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.23085974187.1 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 3 | 5 | 19 |
|---|---|---|---|---|
| Reduction type | ord | nonsplit | split | nonsplit |
| $\lambda$-invariant(s) | 1 | 2 | 1 | 0 |
| $\mu$-invariant(s) | 1 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.