Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-379414416x-1072687188384\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-379414416xz^2-1072687188384z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-30732567723x-782081158035078\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-15935, 962676)$ | $6.2465669314471198017418645859$ | $\infty$ |
| $(-17871, 0)$ | $0$ | $2$ |
Integral points
\( \left(-17871, 0\right) \), \((-15935,\pm 962676)\)
Invariants
| Conductor: | $N$ | = | \( 493680 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $2998396423421341294328064000$ | = | $2^{11} \cdot 3^{28} \cdot 5^{3} \cdot 11^{6} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{1664865424893526702418}{826424127435466125} \) | = | $2 \cdot 3^{-28} \cdot 5^{-3} \cdot 11^{3} \cdot 13^{3} \cdot 17^{-2} \cdot 157^{3} \cdot 419^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.9653106434514083563505943722$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.1309780915389398840204931385$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0609933840271915$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.406403735107465$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.2465669314471198017418645859$ |
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| Real period: | $\Omega$ | ≈ | $0.036002405527380069937041248296$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.5982629731181015737411440028 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $4$ = $2^2$ (rounded) |
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BSD formula
$$\begin{aligned} 3.598262973 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.036002 \cdot 6.246567 \cdot 16}{2^2} \\ & \approx 3.598262973\end{aligned}$$
Modular invariants
Modular form 493680.2.a.e
For more coefficients, see the Downloads section to the right.
| Modular degree: | 309657600 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{3}^{*}$ | additive | 1 | 4 | 11 | 0 |
| $3$ | $2$ | $I_{28}$ | nonsplit multiplicative | 1 | 1 | 28 | 28 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $11$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $17$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 408 & 473 \\ 187 & 1134 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 881 & 968 \\ 44 & 1233 \end{array}\right),\left(\begin{array}{rr} 1313 & 8 \\ 1312 & 9 \end{array}\right),\left(\begin{array}{rr} 56 & 363 \\ 605 & 122 \end{array}\right),\left(\begin{array}{rr} 637 & 44 \\ 1078 & 219 \end{array}\right),\left(\begin{array}{rr} 119 & 0 \\ 0 & 1319 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1314 & 1315 \end{array}\right)$.
The torsion field $K:=\Q(E[1320])$ is a degree-$9732096000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1320\Z)$.
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 | $4$ | \( 605 = 5 \cdot 11^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 32912 = 2^{4} \cdot 11^{2} \cdot 17 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 98736 = 2^{4} \cdot 3 \cdot 11^{2} \cdot 17 \) |
| $7$ | good | $2$ | \( 164560 = 2^{4} \cdot 5 \cdot 11^{2} \cdot 17 \) |
| $11$ | additive | $62$ | \( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \) |
| $17$ | 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 and 4.
Its isogeny class 493680.e
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2040.h2, 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.