Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2+47416x-1450380\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z+47416xz^2-1450380z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+3840669x-1068849054\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(39, 678)$ | $5.4349367328002779998297236681$ | $\infty$ |
| $(30, 0)$ | $0$ | $2$ |
Integral points
\( \left(30, 0\right) \), \((39,\pm 678)\)
Invariants
| Conductor: | $N$ | = | \( 439824 \) | = | $2^{4} \cdot 3 \cdot 7^{2} \cdot 11 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-7751201684717568$ | = | $-1 \cdot 2^{24} \cdot 3 \cdot 7^{7} \cdot 11 \cdot 17 $ |
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| j-invariant: | $j$ | = | \( \frac{24464768327}{16084992} \) | = | $2^{-12} \cdot 3^{-1} \cdot 7^{-1} \cdot 11^{-1} \cdot 17^{-1} \cdot 2903^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7377069782993802457779309328$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.071604723211778283808022439620$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8617725193907065$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.3795049921493265$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.4349367328002779998297236681$ |
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| Real period: | $\Omega$ | ≈ | $0.23738642323712759265696559918$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot1\cdot2^{2}\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.1607207660781529259080660673 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.160720766 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.237386 \cdot 5.434937 \cdot 16}{2^2} \\ & \approx 5.160720766\end{aligned}$$
Modular invariants
Modular form 439824.2.a.dt
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3096576 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{16}^{*}$ | additive | -1 | 4 | 24 | 12 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $17$ | $1$ | $I_{1}$ | nonsplit 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 31416 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 31410 & 31411 \end{array}\right),\left(\begin{array}{rr} 20948 & 1 \\ 10495 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 22180 & 1 \\ 1871 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 25712 & 3 \\ 19997 & 2 \end{array}\right),\left(\begin{array}{rr} 22432 & 31413 \\ 17947 & 31414 \end{array}\right),\left(\begin{array}{rr} 19643 & 19638 \\ 11786 & 27491 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 3919 & 3924 \\ 3922 & 19633 \end{array}\right),\left(\begin{array}{rr} 31409 & 8 \\ 31408 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[31416])$ is a degree-$3201968583475200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/31416\Z)$.
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 | $2$ | \( 27489 = 3 \cdot 7^{2} \cdot 11 \cdot 17 \) |
| $3$ | split multiplicative | $4$ | \( 146608 = 2^{4} \cdot 7^{2} \cdot 11 \cdot 17 \) |
| $7$ | additive | $32$ | \( 8976 = 2^{4} \cdot 3 \cdot 11 \cdot 17 \) |
| $11$ | split multiplicative | $12$ | \( 39984 = 2^{4} \cdot 3 \cdot 7^{2} \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 25872 = 2^{4} \cdot 3 \cdot 7^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 439824.dt
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 7854.i4, its twist by $28$.
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.