Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-643952x-198682560\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-643952xz^2-198682560z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-52160139x-144996066630\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1562, 51030\right) \) | $3.4256653996177597816756480337$ | $\infty$ |
| \( \left(-463, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1562:51030:1]\) | $3.4256653996177597816756480337$ | $\infty$ |
| \([-463:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(14055, 1377810\right) \) | $3.4256653996177597816756480337$ | $\infty$ |
| \( \left(-4170, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-463, 0\right) \), \((1562,\pm 51030)\)
\([-463:0:1]\), \([1562:\pm 51030:1]\)
\( \left(-463, 0\right) \), \((1562,\pm 51030)\)
Invariants
| Conductor: | $N$ | = | \( 3696 \) | = | $2^{4} \cdot 3 \cdot 7 \cdot 11$ |
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| Minimal Discriminant: | $\Delta$ | = | $202790117376$ | = | $2^{13} \cdot 3^{8} \cdot 7^{3} \cdot 11 $ |
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| j-invariant: | $j$ | = | \( \frac{7209828390823479793}{49509306} \) | = | $2^{-1} \cdot 3^{-8} \cdot 7^{-3} \cdot 11^{-1} \cdot 131^{3} \cdot 14747^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7693774599218057626664635931$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0762302793618604532492314716$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0260259228043793$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.298198718839307$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.4256653996177597816756480337$ |
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| Real period: | $\Omega$ | ≈ | $0.16853345518734856591127612940$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 2^{2}\cdot2\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.4640353566799814116566516126 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.464035357 \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 0.168533 \cdot 3.425665 \cdot 24}{2^2} \\ & \approx 3.464035357\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18432 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 4 | 13 | 1 |
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $11$ | $1$ | $I_{1}$ | nonsplit 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 | 8.12.0.6 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 616 = 2^{3} \cdot 7 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 360 & 3 \\ 533 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 69 & 76 \\ 30 & 379 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 228 & 1 \\ 79 & 6 \end{array}\right),\left(\begin{array}{rr} 80 & 393 \\ 531 & 518 \end{array}\right),\left(\begin{array}{rr} 609 & 8 \\ 608 & 9 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 610 & 611 \end{array}\right)$.
The torsion field $K:=\Q(E[616])$ is a degree-$851558400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/616\Z)$.
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 | $4$ | \( 77 = 7 \cdot 11 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 176 = 2^{4} \cdot 11 \) |
| $7$ | split multiplicative | $8$ | \( 528 = 2^{4} \cdot 3 \cdot 11 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 3696q
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 462f4, its twist by $-4$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{154}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-22}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{-22})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.30184.2 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.7055355940864.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.2655855848448.3 | \(\Z/6\Z\) | 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 | \(\Z/16\Z\) | 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 | add | nonsplit | ord | split | nonsplit | ord | ord | ss | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | 1 | 2 | 1 | 1 | 1 | 1,1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 0 | 0 | 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.