Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-1404x-20304\)
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(homogenize, simplify) |
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\(y^2z=x^3-1404xz^2-20304z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1404x-20304\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(148, 1736)$ | $4.4405572421503637594982533500$ | $\infty$ |
Integral points
\((148,\pm 1736)\)
Invariants
| Conductor: | $N$ | = | \( 5184 \) | = | $2^{6} \cdot 3^{4}$ |
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| Discriminant: | $\Delta$ | = | $-967458816$ | = | $-1 \cdot 2^{14} \cdot 3^{10} $ |
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| j-invariant: | $j$ | = | \( -316368 \) | = | $-1 \cdot 2^{4} \cdot 3^{2} \cdot 13^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.59132071722736228183210392325$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1328612339826653219840379159$ |
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| $abc$ quality: | $Q$ | ≈ | $0.938275851490884$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9002684555126956$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.4405572421503637594982533500$ |
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| Real period: | $\Omega$ | ≈ | $0.38989312857958032521947012886$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.4626855115574367661795898118 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.462685512 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.389893 \cdot 4.440557 \cdot 2}{1^2} \\ & \approx 3.462685512\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3456 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}^{*}$ | additive | 1 | 6 | 14 | 0 |
| $3$ | $1$ | $IV^{*}$ | additive | 1 | 4 | 10 | 0 |
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 |
|---|---|---|
| $2$ | 2G | 4.2.0.1 |
| $3$ | 3B | 3.4.0.1 |
| $5$ | 5S4 | 5.5.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $160$, genus $4$, and generators
$\left(\begin{array}{rr} 61 & 105 \\ 90 & 61 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 60 & 1 \end{array}\right),\left(\begin{array}{rr} 61 & 66 \\ 78 & 49 \end{array}\right),\left(\begin{array}{rr} 88 & 69 \\ 27 & 103 \end{array}\right),\left(\begin{array}{rr} 61 & 60 \\ 60 & 61 \end{array}\right),\left(\begin{array}{rr} 109 & 42 \\ 66 & 97 \end{array}\right),\left(\begin{array}{rr} 1 & 60 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 39 & 20 \\ 80 & 39 \end{array}\right),\left(\begin{array}{rr} 101 & 20 \\ 100 & 21 \end{array}\right),\left(\begin{array}{rr} 59 & 60 \\ 30 & 119 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 97 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$221184$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
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 | $2$ | \( 81 = 3^{4} \) |
| $3$ | additive | $4$ | \( 32 = 2^{5} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 5184l
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 324d1, its twist by $-24$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.324.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.419904.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.120932352.6 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.40310784.1 | \(\Z/6\Z\) | not in database |
| $12$ | 12.2.1624959306694656.3 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.131621703842267136.93 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | 12.0.6499837226778624.45 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.34811183694905557468330328064.3 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.185659646372829639831095083008.2 | \(\Z/6\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | - | 1 | 1 | 1 | 5 | 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 |
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.