Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-71370x+8011575\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-71370xz^2+8011575z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1141923x+511598878\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{519}{4}, \frac{7167}{8}\right) \) | $5.5448552134147450120533122127$ | $\infty$ |
| \( \left(-\frac{1245}{4}, \frac{1245}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1038:7167:8]\) | $5.5448552134147450120533122127$ | $\infty$ |
| \([-2490:1245:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(518, 7686\right) \) | $5.5448552134147450120533122127$ | $\infty$ |
| \( \left(-1246, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 585 \) | = | $3^{2} \cdot 5 \cdot 13$ |
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| Minimal Discriminant: | $\Delta$ | = | $-4338226318359375$ | = | $-1 \cdot 3^{7} \cdot 5^{16} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{55150149867714721}{5950927734375} \) | = | $-1 \cdot 3^{-1} \cdot 5^{-16} \cdot 13^{-1} \cdot 380641^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7387232121286887326718194920$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1894170677946338869741968735$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0142537292135114$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.111463053052766$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.5448552134147450120533122127$ |
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| Real period: | $\Omega$ | ≈ | $0.42554072646310054843282797780$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.3595617156492210218470390653 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.359561716 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.425541 \cdot 5.544855 \cdot 4}{2^2} \\ & \approx 2.359561716\end{aligned}$$
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 not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $2$ | $I_{16}$ | nonsplit multiplicative | 1 | 1 | 16 | 16 |
| $13$ | $1$ | $I_{1}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 16.48.0.110 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 1951 & 32 \\ 406 & 513 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 2497 & 32 \\ 2512 & 513 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 6210 & 6209 \\ 2231 & 156 \end{array}\right),\left(\begin{array}{rr} 6209 & 32 \\ 6208 & 33 \end{array}\right),\left(\begin{array}{rr} 2341 & 32 \\ 2356 & 513 \end{array}\right),\left(\begin{array}{rr} 1942 & 3 \\ 2285 & 6028 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 3678 & 4235 \end{array}\right)$.
The torsion field $K:=\Q(E[6240])$ is a degree-$309162147840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6240\Z)$.
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 | $2$ | \( 117 = 3^{2} \cdot 13 \) |
| $3$ | additive | $8$ | \( 65 = 5 \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 117 = 3^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 45 = 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 585.g
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 195.a3, its twist by $-3$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.0.39.1-975.2-e1 |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-13}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{-13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{26})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{3})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.56299900176.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.230604391120896.5 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.151613669376.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.8.6559413791883264.4 | \(\Z/16\Z\) | not in database |
| $8$ | 8.0.6559413791883264.52 | \(\Z/16\Z\) | not in database |
| $8$ | 8.2.3162184666875.6 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/16\Z\) | not in database |
| $16$ | 16.0.43025909293148379807167803293696.10 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ord | add | nonsplit | ss | ord | split | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | - | 1 | 1,1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 1 |
| $\mu$-invariant(s) | 1 | - | 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.