Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-6221800x-5975999000\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-6221800xz^2-5975999000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-8063453475x-278695257545250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(3115, 68305)$ | $4.6174351836505479382630075861$ | $\infty$ |
| $(-5765/4, 5765/8)$ | $0$ | $2$ |
Integral points
\( \left(3115, 68305\right) \), \( \left(3115, -71420\right) \)
Invariants
| Conductor: | $N$ | = | \( 124950 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $11481939448875000$ | = | $2^{3} \cdot 3^{8} \cdot 5^{6} \cdot 7^{7} \cdot 17 $ |
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| j-invariant: | $j$ | = | \( \frac{14489843500598257}{6246072} \) | = | $2^{-3} \cdot 3^{-8} \cdot 7^{-1} \cdot 11^{3} \cdot 17^{-1} \cdot 37^{3} \cdot 599^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4242239012173689033801281991$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.64654987047266206352707216076$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9901853459055785$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.9885790948574975$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.6174351836505479382630075861$ |
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| Real period: | $\Omega$ | ≈ | $0.095591779243311143684940784519$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.5311107579665682173475397891 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $4$ = $2^2$ (rounded) |
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BSD formula
$$\begin{aligned} 3.531110758 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.095592 \cdot 4.617435 \cdot 8}{2^2} \\ & \approx 3.531110758\end{aligned}$$
Modular invariants
Modular form 124950.2.a.bb
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4718592 |
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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 |
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $17$ | $1$ | $I_{1}$ | split 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 | 8.12.0.9 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4760 = 2^{3} \cdot 5 \cdot 7 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1236 & 1905 \\ 3775 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1903 & 0 \\ 0 & 4759 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 4754 & 4755 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 4753 & 8 \\ 4752 & 9 \end{array}\right),\left(\begin{array}{rr} 3124 & 2855 \\ 1065 & 4754 \end{array}\right),\left(\begin{array}{rr} 2976 & 1555 \\ 2265 & 3246 \end{array}\right),\left(\begin{array}{rr} 3216 & 2745 \\ 385 & 2336 \end{array}\right)$.
The torsion field $K:=\Q(E[4760])$ is a degree-$2425733775360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4760\Z)$.
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 | $4$ | \( 20825 = 5^{2} \cdot 7^{2} \cdot 17 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 20825 = 5^{2} \cdot 7^{2} \cdot 17 \) |
| $5$ | additive | $14$ | \( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \) |
| $7$ | additive | $32$ | \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 124950t
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 714f4, its twist by $-35$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{238}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-85}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-70}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-70}, \sqrt{-85})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | nonsplit | add | add | ss | ord | split | ss | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 9 | 1 | - | - | 1,1 | 1 | 2 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 1 | 0 | - | - | 0,0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.