Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-83974076x-296243229202\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-83974076xz^2-296243229202z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-108830401875x-13821197610431250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(10713, 178177)$ | $6.0366606085138256197298416964$ | $\infty$ |
Integral points
\( \left(10713, 178177\right) \), \( \left(10713, -188891\right) \)
Invariants
| Conductor: | $N$ | = | \( 481650 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 13^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $-12605263176892851562500$ | = | $-1 \cdot 2^{2} \cdot 3^{3} \cdot 5^{10} \cdot 13^{6} \cdot 19^{5} $ |
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| j-invariant: | $j$ | = | \( -\frac{1389310279182025}{267418692} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-3} \cdot 5^{2} \cdot 19^{-5} \cdot 31^{3} \cdot 1231^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2411399389478999331588392789$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.61746699985538125296479611376$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0156937260858383$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.070854307557088$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.0366606085138256197298416964$ |
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| Real period: | $\Omega$ | ≈ | $0.024936083074859953710966963756$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 60 $ = $ 2\cdot3\cdot1\cdot2\cdot5 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $9.0318402257181237613963451954 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.031840226 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.024936 \cdot 6.036661 \cdot 60}{1^2} \\ & \approx 9.031840226\end{aligned}$$
Modular invariants
Modular form 481650.2.a.dw
For more coefficients, see the Downloads section to the right.
| Modular degree: | 69120000 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $1$ | $II^{*}$ | additive | 1 | 2 | 10 | 0 |
| $13$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $19$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
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 |
|---|---|---|
| $5$ | 5B.4.2 | 5.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 14820 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \cdot 19 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 14811 & 10 \\ 14810 & 11 \end{array}\right),\left(\begin{array}{rr} 14247 & 4550 \\ 9100 & 1871 \end{array}\right),\left(\begin{array}{rr} 12546 & 12545 \\ 9685 & 2276 \end{array}\right),\left(\begin{array}{rr} 2406 & 12545 \\ 13195 & 2276 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 12546 & 12545 \\ 12155 & 2276 \end{array}\right),\left(\begin{array}{rr} 11399 & 0 \\ 0 & 14819 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 14765 & 14701 \end{array}\right)$.
The torsion field $K:=\Q(E[14820])$ is a degree-$148687670476800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/14820\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 240825 = 3 \cdot 5^{2} \cdot 13^{2} \cdot 19 \) |
| $3$ | split multiplicative | $4$ | \( 160550 = 2 \cdot 5^{2} \cdot 13^{2} \cdot 19 \) |
| $5$ | additive | $2$ | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 25350 = 2 \cdot 3 \cdot 5^{2} \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 481650dw
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 2850g1, its twist by $65$.
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.