Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-2801175x-264189250\)
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(homogenize, simplify) |
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\(y^2z=x^3-2801175xz^2-264189250z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2801175x-264189250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{19423892030}{2163841}, \frac{2659127484481850}{3183010111}\right) \) | $22.164314607592293904188232528$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([28572545176130:2659127484481850:3183010111]\) | $22.164314607592293904188232528$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{19423892030}{2163841}, \frac{2659127484481850}{3183010111}\right) \) | $22.164314607592293904188232528$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 456300 \) | = | $2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 13^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $1376545591687500000000$ | = | $2^{8} \cdot 3^{3} \cdot 5^{12} \cdot 13^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{27591408}{15625} \) | = | $2^{4} \cdot 3^{3} \cdot 5^{-6} \cdot 13 \cdot 17^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7461129566094675456260293121$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.50532343045559809483697470576$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9781573886709247$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.309006037417324$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $22.164314607592293904188232528$ |
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| Real period: | $\Omega$ | ≈ | $0.12588321367105520895868220734$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.5802303032400618422466961695 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.580230303 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.125883 \cdot 22.164315 \cdot 2}{1^2} \\ & \approx 5.580230303\end{aligned}$$
Modular invariants
Modular form 456300.2.a.be
For more coefficients, see the Downloads section to the right.
| Modular degree: | 19408896 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $3$ | $1$ | $II$ | additive | -1 | 3 | 3 | 0 |
| $5$ | $2$ | $I_{6}^{*}$ | additive | 1 | 2 | 12 | 6 |
| $13$ | $1$ | $IV^{*}$ | additive | 1 | 2 | 8 | 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 | $\ell$-adic index |
|---|---|---|---|
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 60.16.0-12.b.1.1, level \( 60 = 2^{2} \cdot 3 \cdot 5 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 6 & 25 \\ 55 & 6 \end{array}\right),\left(\begin{array}{rr} 9 & 10 \\ 50 & 29 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 23 & 0 \\ 0 & 59 \end{array}\right),\left(\begin{array}{rr} 55 & 6 \\ 54 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 31 & 30 \\ 45 & 31 \end{array}\right)$.
The torsion field $K:=\Q(E[60])$ is a degree-$138240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/60\Z)$.
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 | $2$ | \( 114075 = 3^{3} \cdot 5^{2} \cdot 13^{2} \) |
| $3$ | additive | $2$ | \( 16900 = 2^{2} \cdot 5^{2} \cdot 13^{2} \) |
| $5$ | additive | $18$ | \( 18252 = 2^{2} \cdot 3^{3} \cdot 13^{2} \) |
| $13$ | additive | $74$ | \( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 456300be
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 91260e1, its twist by $65$.
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.