Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-1875x+59875\)
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(homogenize, simplify) |
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\(y^2z=x^3-1875xz^2+59875z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1875x+59875\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(5, 225)$ | $1.0162279602023949643176192667$ | $\infty$ |
| $(30, 175)$ | $2.4269200184166441981489400552$ | $\infty$ |
Integral points
\((-31,\pm 297)\), \((5,\pm 225)\), \((6,\pm 221)\), \((30,\pm 175)\), \((221,\pm 3231)\), \((198905,\pm 88709175)\)
Invariants
| Conductor: | $N$ | = | \( 412200 \) | = | $2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 229$ |
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| Discriminant: | $\Delta$ | = | $-1126851750000$ | = | $-1 \cdot 2^{4} \cdot 3^{9} \cdot 5^{6} \cdot 229 $ |
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| j-invariant: | $j$ | = | \( -\frac{4000000}{6183} \) | = | $-1 \cdot 2^{8} \cdot 3^{-3} \cdot 5^{6} \cdot 229^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0039806323360602111752119158$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.58109352840169325829520107643$ |
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| $abc$ quality: | $Q$ | ≈ | $0.822530403855858$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.7475016637797296$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.3797963755440370133751548327$ |
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| Real period: | $\Omega$ | ≈ | $0.78054922999785116534510997558$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $14.860385827860601063665008833 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.860385828 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.780549 \cdot 2.379796 \cdot 8}{1^2} \\ & \approx 14.860385828\end{aligned}$$
Modular invariants
Modular form 412200.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 442368 |
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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 4 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 |
| $3$ | $2$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $229$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1374 = 2 \cdot 3 \cdot 229 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1373 & 2 \\ 1372 & 3 \end{array}\right),\left(\begin{array}{rr} 235 & 2 \\ 235 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 917 & 2 \\ 917 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 1373 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[1374])$ is a degree-$394271608320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1374\Z)$.
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 | $2$ | \( 51525 = 3^{2} \cdot 5^{2} \cdot 229 \) |
| $3$ | additive | $2$ | \( 45800 = 2^{3} \cdot 5^{2} \cdot 229 \) |
| $5$ | additive | $14$ | \( 16488 = 2^{3} \cdot 3^{2} \cdot 229 \) |
| $229$ | split multiplicative | $230$ | \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 412200.h consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 5496.a1, its twist by $-15$.
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.