Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-42574818x+106904677633\)
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(homogenize, simplify) |
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\(y^2z=x^3-42574818xz^2+106904677633z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-42574818x+106904677633\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1781, 191646)$ | $1.8580048783664183112901940835$ | $\infty$ |
| $(3809, 0)$ | $0$ | $2$ |
Integral points
\((1781,\pm 191646)\), \( \left(3809, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 425880 \) | = | $2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $1833023585847431250000$ | = | $2^{4} \cdot 3^{11} \cdot 5^{8} \cdot 7^{3} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{151591373397612544}{32558203125} \) | = | $2^{11} \cdot 3^{-5} \cdot 5^{-8} \cdot 7^{-3} \cdot 11^{6} \cdot 347^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0748701538163403226619725978$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0120402705648686724651955514$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1312171453358946$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.961761746176395$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.8580048783664183112901940835$ |
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| Real period: | $\Omega$ | ≈ | $0.14442579226939676780900309861$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2\cdot2^{2}\cdot2\cdot3\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.4402518383633793507998846546 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.440251838 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.144426 \cdot 1.858005 \cdot 96}{2^2} \\ & \approx 6.440251838\end{aligned}$$
Modular invariants
Modular form 425880.2.a.cc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 29491200 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | 1 | 3 | 4 | 0 |
| $3$ | $4$ | $I_{5}^{*}$ | additive | -1 | 2 | 11 | 5 |
| $5$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $13$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2184 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 2178 & 2179 \end{array}\right),\left(\begin{array}{rr} 2177 & 8 \\ 2176 & 9 \end{array}\right),\left(\begin{array}{rr} 1457 & 780 \\ 1118 & 1535 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 167 & 0 \\ 0 & 2183 \end{array}\right),\left(\begin{array}{rr} 1132 & 169 \\ 2015 & 846 \end{array}\right),\left(\begin{array}{rr} 272 & 1677 \\ 611 & 2014 \end{array}\right),\left(\begin{array}{rr} 235 & 234 \\ 1378 & 1795 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[2184])$ is a degree-$81155063808$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2184\Z)$.
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 | $2$ | \( 10647 = 3^{2} \cdot 7 \cdot 13^{2} \) |
| $3$ | additive | $8$ | \( 6760 = 2^{3} \cdot 5 \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 85176 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13^{2} \) |
| $7$ | split multiplicative | $8$ | \( 60840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 425880cc
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 840d1, its twist by $-39$.
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.