Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-334425x+188778625\)
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(homogenize, simplify) |
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\(y^2z=x^3-334425xz^2+188778625z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-334425x+188778625\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(245, 11025)$ | $0.97746762018894834588243579573$ | $\infty$ |
| $(-735, 6125)$ | $1.1975368089433410608438515956$ | $\infty$ |
Integral points
\((-735,\pm 6125)\), \((-441,\pm 15827)\), \((29,\pm 13383)\), \((245,\pm 11025)\), \((515,\pm 12375)\), \((2265,\pm 105125)\), \((12495,\pm 1395275)\)
Invariants
| Conductor: | $N$ | = | \( 485100 \) | = | $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $-13001607905343750000$ | = | $-1 \cdot 2^{4} \cdot 3^{8} \cdot 5^{9} \cdot 7^{8} \cdot 11 $ |
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| j-invariant: | $j$ | = | \( -\frac{3937024}{12375} \) | = | $-1 \cdot 2^{8} \cdot 3^{-2} \cdot 5^{-3} \cdot 7 \cdot 11^{-1} \cdot 13^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3541829636256159986458488291$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.52816462981567967422813265876$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7692609731071308$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9440025633636653$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.1524234147229023824674247917$ |
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| Real period: | $\Omega$ | ≈ | $0.19699289109234915608834428794$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 72 $ = $ 3\cdot2\cdot2^{2}\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $16.345383856472292050819070570 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 16.345383856 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.196993 \cdot 1.152423 \cdot 72}{1^2} \\ & \approx 16.345383856\end{aligned}$$
Modular invariants
Modular form 485100.2.a.do
For more coefficients, see the Downloads section to the right.
| Modular degree: | 9289728 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $4$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $7$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
| $11$ | $1$ | $I_{1}$ | 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 \( 110 = 2 \cdot 5 \cdot 11 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 109 & 0 \end{array}\right),\left(\begin{array}{rr} 109 & 2 \\ 108 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 67 & 2 \\ 67 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 101 & 2 \\ 101 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[110])$ is a degree-$19008000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/110\Z)$.
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 | $2$ | \( 121275 = 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11 \) |
| $3$ | additive | $8$ | \( 53900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11 \) |
| $5$ | additive | $18$ | \( 19404 = 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11 \) |
| $7$ | additive | $26$ | \( 9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 44100 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 485100.do consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 32340.bi1, its twist by $105$.
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.