Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-3150622240x+67341792313600\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-3150622240xz^2+67341792313600z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-255200401467x+49091400995410026\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(24480, 2210000)$ | $1.9547873100534137018884617469$ | $\infty$ |
| $(11730, 5656750)$ | $0$ | $4$ |
Integral points
\((-16320,\pm 10696400)\), \((11730,\pm 5656750)\), \((24480,\pm 2210000)\), \( \left(35105, 0\right) \), \((344480,\pm 199650000)\)
Invariants
| Conductor: | $N$ | = | \( 493680 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $42538640205602376000000000000$ | = | $2^{15} \cdot 3^{3} \cdot 5^{12} \cdot 11^{9} \cdot 17^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{476646772170172569823801}{5862293314453125000} \) | = | $2^{-3} \cdot 3^{-3} \cdot 5^{-12} \cdot 11^{-3} \cdot 17^{-4} \cdot 23^{3} \cdot 107^{3} \cdot 31741^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.3035761215371378145650002670$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.4114813045780072331167963566$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0086073801975677$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.890793636028198$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.9547873100534137018884617469$ |
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| Real period: | $\Omega$ | ≈ | $0.036257494305358402820513720813$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 768 $ = $ 2^{2}\cdot1\cdot( 2^{2} \cdot 3 )\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.4020331085975288557179572295 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.402033109 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.036257 \cdot 1.954787 \cdot 768}{4^2} \\ & \approx 3.402033109\end{aligned}$$
Modular invariants
Modular form 493680.2.a.cf
For more coefficients, see the Downloads section to the right.
| Modular degree: | 637009920 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{7}^{*}$ | additive | -1 | 4 | 15 | 3 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $5$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $11$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $17$ | $4$ | $I_{4}$ | 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 |
|---|---|---|
| $2$ | 2B | 4.12.0.7 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 22440 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 17 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 14521 & 24 \\ 17172 & 289 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 21134 & 13211 \end{array}\right),\left(\begin{array}{rr} 12162 & 15919 \\ 7037 & 11972 \end{array}\right),\left(\begin{array}{rr} 16 & 11241 \\ 3455 & 7106 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 20384 & 22437 \\ 12699 & 86 \end{array}\right),\left(\begin{array}{rr} 22417 & 24 \\ 22416 & 25 \end{array}\right),\left(\begin{array}{rr} 5609 & 22416 \\ 5610 & 22439 \end{array}\right),\left(\begin{array}{rr} 8977 & 24 \\ 17964 & 289 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[22440])$ is a degree-$95296684032000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/22440\Z)$.
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 | $4$ | \( 363 = 3 \cdot 11^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 32912 = 2^{4} \cdot 11^{2} \cdot 17 \) |
| $5$ | split multiplicative | $6$ | \( 98736 = 2^{4} \cdot 3 \cdot 11^{2} \cdot 17 \) |
| $11$ | additive | $72$ | \( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 29040 = 2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 493680.cf
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 5610.q5, its twist by $44$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.