Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-5611520x-5051181132\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-5611520xz^2-5051181132z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-454533147x-3680947445814\)
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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 |
|---|---|---|
| $(-1238, 0)$ | $0$ | $2$ |
| $(-1364, 8190)$ | $0$ | $4$ |
Integral points
\( \left(-1494, 0\right) \), \((-1364,\pm 8190)\), \( \left(-1238, 0\right) \), \( \left(2731, 0\right) \), \((6826,\pm 524160)\)
Invariants
| Conductor: | $N$ | = | \( 21840 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $294859780339138560000$ | = | $2^{20} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4} \cdot 13^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{4770955732122964500481}{71987251059360000} \) | = | $2^{-8} \cdot 3^{-8} \cdot 5^{-4} \cdot 7^{-4} \cdot 13^{-4} \cdot 3217^{3} \cdot 5233^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7299982922304362136456477578$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.0368511116704909042284156363$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0084981242776025$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.828415044686723$ | |||
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: | $\Omega$ | ≈ | $0.098180763360164677398985313354$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2048 $ = $ 2^{2}\cdot2^{3}\cdot2^{2}\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $8$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.1417844275252696767675300273 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 3.141784428 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.098181 \cdot 1.000000 \cdot 2048}{8^2} \\ & \approx 3.141784428\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 786432 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{12}^{*}$ | additive | -1 | 4 | 20 | 8 |
| $3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $7$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $13$ | $4$ | $I_{4}$ | split 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 |
|---|---|---|
| $2$ | 2Cs | 8.96.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7280 = 2^{4} \cdot 5 \cdot 7 \cdot 13 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 6385 & 8 \\ 328 & 4725 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 8 & 129 \end{array}\right),\left(\begin{array}{rr} 895 & 7204 \\ 376 & 1905 \end{array}\right),\left(\begin{array}{rr} 4381 & 12 \\ 3004 & 85 \end{array}\right),\left(\begin{array}{rr} 2805 & 4 \\ 4436 & 7245 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 4161 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7265 & 16 \\ 7264 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[7280])$ is a degree-$811550638080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7280\Z)$.
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 | $2$ | \( 1 \) |
| $3$ | split multiplicative | $4$ | \( 7280 = 2^{4} \cdot 5 \cdot 7 \cdot 13 \) |
| $5$ | split multiplicative | $6$ | \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 21840.cg
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 2730.w4, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{14}, \sqrt{130})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-14}, \sqrt{130})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{65})\) | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.157351936.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.2808830402560000.112 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | 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 | 13 |
|---|---|---|---|---|---|
| Reduction type | add | split | split | split | split |
| $\lambda$-invariant(s) | - | 1 | 1 | 3 | 1 |
| $\mu$-invariant(s) | - | 0 | 0 | 0 | 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 $0$.