Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-37633153x-88862666644\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-37633153xz^2-88862666644z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-48772565667x-4145830257233826\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 424830 \) | = | $2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-288889166592600$ | = | $-1 \cdot 2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{8} \cdot 17^{4} $ |
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| j-invariant: | $j$ | = | \( -\frac{12242088317612041}{600} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-1} \cdot 5^{-2} \cdot 7 \cdot 17^{2} \cdot 18223^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6984477745554089595995201483$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.45676989383312806277944011338$ |
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| $abc$ quality: | $Q$ | ≈ | $1.014544776657247$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.934146019318314$ | |||
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: | $\Omega$ | ≈ | $0.030477329394755357422389430642$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 1\cdot1\cdot2\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.6457757873167893008090292547 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 1.645775787 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.030477 \cdot 1.000000 \cdot 6}{1^2} \\ & \approx 1.645775787\end{aligned}$$
Modular invariants
Modular form 424830.2.a.dt
For more coefficients, see the Downloads section to the right.
| Modular degree: | 20575296 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
| $17$ | $1$ | $IV$ | additive | -1 | 2 | 4 | 0 |
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 |
|---|---|---|
| $3$ | 3B.1.2 | 3.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 24.16.0-24.d.1.7, level \( 24 = 2^{3} \cdot 3 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 2 & 3 \\ 1 & 10 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 22 \\ 14 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 6 \\ 15 & 19 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 19 & 6 \\ 18 & 7 \end{array}\right)$.
The torsion field $K:=\Q(E[24])$ is a degree-$4608$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24\Z)$.
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 | $4$ | \( 42483 = 3 \cdot 7^{2} \cdot 17^{2} \) |
| $3$ | split multiplicative | $4$ | \( 70805 = 5 \cdot 7^{2} \cdot 17^{2} \) |
| $5$ | split multiplicative | $6$ | \( 84966 = 2 \cdot 3 \cdot 7^{2} \cdot 17^{2} \) |
| $7$ | additive | $26$ | \( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \) |
| $17$ | additive | $114$ | \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 424830dt
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 424830g2, its twist by $-7$.
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 $0$.