Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-14807415x+21922426638\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-14807415xz^2+21922426638z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1199400642x+15985047221001\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2121, 7875)$ | $0.64840732713103725984607047048$ | $\infty$ |
| $(2246, 0)$ | $0$ | $2$ |
Integral points
\((-1959,\pm 208365)\), \((-399,\pm 166635)\), \((2121,\pm 7875)\), \( \left(2246, 0\right) \), \((2646,\pm 35700)\)
Invariants
| Conductor: | $N$ | = | \( 444360 \) | = | $2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $77116520701631250000$ | = | $2^{4} \cdot 3^{5} \cdot 5^{8} \cdot 7^{3} \cdot 23^{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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8108364387160919543409826745$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0120402705648686724651955514$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1312171453358946$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.701913542935569$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.64840732713103725984607047048$ |
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| Real period: | $\Omega$ | ≈ | $0.18806723698593808338277183658$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 480 $ = $ 2\cdot5\cdot2^{3}\cdot3\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $14.633300934596575756222210569 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.633300935 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.188067 \cdot 0.648407 \cdot 480}{2^2} \\ & \approx 14.633300935\end{aligned}$$
Modular invariants
Modular form 444360.2.a.cq
For more coefficients, see the Downloads section to the right.
| Modular degree: | 23654400 |
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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 |
| $3$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $5$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $23$ | $2$ | $I_0^{*}$ | 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 \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3857 & 8 \\ 3856 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1243 & 1242 \\ 874 & 1795 \end{array}\right),\left(\begin{array}{rr} 2416 & 2691 \\ 1909 & 2186 \end{array}\right),\left(\begin{array}{rr} 1007 & 0 \\ 0 & 3863 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3858 & 3859 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 1289 & 276 \\ 782 & 1703 \end{array}\right),\left(\begin{array}{rr} 2140 & 2185 \\ 1679 & 3198 \end{array}\right)$.
The torsion field $K:=\Q(E[3864])$ is a degree-$827306016768$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3864\Z)$.
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 | $2$ | \( 11109 = 3 \cdot 7 \cdot 23^{2} \) |
| $3$ | split multiplicative | $4$ | \( 21160 = 2^{3} \cdot 5 \cdot 23^{2} \) |
| $5$ | split multiplicative | $6$ | \( 29624 = 2^{3} \cdot 7 \cdot 23^{2} \) |
| $7$ | split multiplicative | $8$ | \( 63480 = 2^{3} \cdot 3 \cdot 5 \cdot 23^{2} \) |
| $23$ | additive | $266$ | \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 444360.cq
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 840.h3, its twist by $-23$.
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.