Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-6170x+54695\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-6170xz^2+54695z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7996347x+2671803414\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(9, -5\right) \) | $0$ | $2$ |
| \( \left(-27, 463\right) \) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([9:-5:1]\) | $0$ | $2$ |
| \([-27:463:1]\) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(339, 0\right) \) | $0$ | $2$ |
| \( \left(-957, 97200\right) \) | $0$ | $4$ |
Integral points
\( \left(-27, 463\right) \), \( \left(-27, -437\right) \), \( \left(9, -5\right) \), \( \left(73, -37\right) \), \( \left(173, 1963\right) \), \( \left(173, -2137\right) \)
\([-27:463:1]\), \([-27:-437:1]\), \([9:-5:1]\), \([73:-37:1]\), \([173:1963:1]\), \([173:-2137:1]\)
\((-957,\pm 97200)\), \( \left(339, 0\right) \), \( \left(2643, 0\right) \), \((6243,\pm 442800)\)
Invariants
| Conductor: | $N$ | = | \( 1230 \) | = | $2 \cdot 3 \cdot 5 \cdot 41$ |
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| Minimal Discriminant: | $\Delta$ | = | $13616100000000$ | = | $2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 41^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{25976677550021281}{13616100000000} \) | = | $2^{-8} \cdot 3^{-4} \cdot 5^{-8} \cdot 41^{-2} \cdot 73^{3} \cdot 4057^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2119152946586460740866265348$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2119152946586460740866265348$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0097650965595288$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.312326110826036$ | |||
| 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: | $\Omega$ | ≈ | $0.62058515969098032724321346158$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 256 $ = $ 2^{3}\cdot2\cdot2^{3}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $8$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.4823406387639213089728538463 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.482340639 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.620585 \cdot 1.000000 \cdot 256}{8^2} \\ & \approx 2.482340639\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3072 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $5$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $41$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 |
|---|---|---|---|
| $2$ | 2Cs | 8.96.0.39 | $96$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3280 = 2^{4} \cdot 5 \cdot 41 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 2633 & 8 \\ 612 & 3241 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 68 & 929 \end{array}\right),\left(\begin{array}{rr} 15 & 8 \\ 448 & 2289 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3265 & 16 \\ 3264 & 17 \end{array}\right),\left(\begin{array}{rr} 645 & 8 \\ 3198 & 3149 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3280])$ is a degree-$42319872000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3280\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 1 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 410 = 2 \cdot 5 \cdot 41 \) |
| $5$ | split multiplicative | $6$ | \( 246 = 2 \cdot 3 \cdot 41 \) |
| $41$ | split multiplicative | $42$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 1230f
consists of 8 curves linked by isogenies of
degrees dividing 16.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $2$ | \(\Q(\sqrt{41}) \) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{41})\) | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.21785760000.6 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | 8.0.12160266856960000.23 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | 8.8.12160266856960000.11 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | 8.2.5005750838670000.8 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.37855255048314838297713049600000000.2 | \(\Z/4\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | 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 | 41 |
|---|---|---|---|---|
| Reduction type | split | nonsplit | split | split |
| $\lambda$-invariant(s) | 2 | 0 | 1 | 1 |
| $\mu$-invariant(s) | 1 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.