Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-2822427075x-57712268722750\)
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(homogenize, simplify) |
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\(y^2z=x^3-2822427075xz^2-57712268722750z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2822427075x-57712268722750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-30815, 0)$ | $0$ | $2$ |
| $(61345, 0)$ | $0$ | $2$ |
Integral points
\( \left(-30815, 0\right) \), \( \left(-30530, 0\right) \), \( \left(61345, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 478800 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $93172841201664000000000000$ | = | $2^{26} \cdot 3^{8} \cdot 5^{12} \cdot 7^{4} \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{53294746224000958661881}{1997017344000000} \) | = | $2^{-14} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{-4} \cdot 13^{3} \cdot 19^{-2} \cdot 79^{3} \cdot 36643^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.0695457686755867590455536055$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.0223734875645364166303191990$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0082885109963067$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.879346064833229$ | |||
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: | $\Omega$ | ≈ | $0.020713087925172211901209476390$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 256 $ = $ 2^{2}\cdot2^{2}\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.3256376272110215616774064889 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 1.325637627 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.020713 \cdot 1.000000 \cdot 256}{4^2} \\ & \approx 1.325637627\end{aligned}$$
Modular invariants
Modular form 478800.2.a.bb
For more coefficients, see the Downloads section to the right.
| Modular degree: | 247726080 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{18}^{*}$ | additive | -1 | 4 | 26 | 14 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $4$ | $I_{6}^{*}$ | additive | 1 | 2 | 12 | 6 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $19$ | $2$ | $I_{2}$ | 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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 909 & 2278 \\ 914 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1921 & 4 \\ 1562 & 9 \end{array}\right),\left(\begin{array}{rr} 2277 & 4 \\ 2276 & 5 \end{array}\right),\left(\begin{array}{rr} 1519 & 2278 \\ 0 & 2279 \end{array}\right),\left(\begin{array}{rr} 569 & 2278 \\ 0 & 2279 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1141 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[2280])$ is a degree-$90773913600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2280\Z)$.
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 | $2$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $3$ | additive | $8$ | \( 53200 = 2^{4} \cdot 5^{2} \cdot 7 \cdot 19 \) |
| $5$ | additive | $18$ | \( 19152 = 2^{4} \cdot 3^{2} \cdot 7 \cdot 19 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 68400 = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 25200 = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 478800.bb
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 3990.g3, its twist by $60$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.