Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+9876028596x-98580268761968\)
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(homogenize, simplify) |
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\(y^2z=x^3+9876028596xz^2-98580268761968z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+9876028596x-98580268761968\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(444055468754488320726506984385657002226848636/32026064313969110245041834161526476747929, 36707742519009023372043397967564113869077344365394154553224957369040/5731329235672321382281697445856521048782112956192606063657533)$ | $103.01856609950557243877922701$ | $\infty$ |
| $(9884, 0)$ | $0$ | $2$ |
Integral points
\( \left(9884, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 479808 \) | = | $2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-65847341385090655872686827241472$ | = | $-1 \cdot 2^{24} \cdot 3^{10} \cdot 7^{22} \cdot 17 $ |
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| j-invariant: | $j$ | = | \( \frac{4738217997934888496063}{2928751705237796928} \) | = | $2^{-6} \cdot 3^{-4} \cdot 7^{-16} \cdot 17^{-1} \cdot 47^{3} \cdot 191^{3} \cdot 1871^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.7945373687446745702023382328$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.2325553790430451078261910604$ |
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| $abc$ quality: | $Q$ | ≈ | $1.06742381122777$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.165649470269178$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $103.01856609950557243877922701$ |
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| Real period: | $\Omega$ | ≈ | $0.011313586968407904361364826045$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $9.3240760554194764640023326362 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.324076055 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.011314 \cdot 103.018566 \cdot 32}{2^2} \\ & \approx 9.324076055\end{aligned}$$
Modular invariants
Modular form 479808.2.a.ni
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1132462080 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{14}^{*}$ | additive | -1 | 6 | 24 | 6 |
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $7$ | $4$ | $I_{16}^{*}$ | additive | -1 | 2 | 22 | 16 |
| $17$ | $1$ | $I_{1}$ | split 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 |
|---|---|---|
| $2$ | 2B | 16.48.0.120 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 2447 & 0 \\ 0 & 5711 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1903 & 0 \\ 0 & 5711 \end{array}\right),\left(\begin{array}{rr} 1408 & 1365 \\ 3843 & 1618 \end{array}\right),\left(\begin{array}{rr} 2435 & 1344 \\ 756 & 1511 \end{array}\right),\left(\begin{array}{rr} 5697 & 16 \\ 5696 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 5614 & 5699 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 5708 & 5709 \end{array}\right),\left(\begin{array}{rr} 3086 & 5229 \\ 5313 & 2834 \end{array}\right)$.
The torsion field $K:=\Q(E[5712])$ is a degree-$970293510144$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5712\Z)$.
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 | $2$ | \( 7497 = 3^{2} \cdot 7^{2} \cdot 17 \) |
| $3$ | additive | $8$ | \( 53312 = 2^{6} \cdot 7^{2} \cdot 17 \) |
| $7$ | additive | $32$ | \( 9792 = 2^{6} \cdot 3^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 28224 = 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 479808.ni
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 714.f6, its twist by $-168$.
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.