Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2+171220x-2571900\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z+171220xz^2-2571900z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+13868793x-1916521506\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(340, 9750)$ | $1.8093134794444246505765552823$ | $\infty$ |
| $(15, 0)$ | $0$ | $2$ |
Integral points
\( \left(15, 0\right) \), \((340,\pm 9750)\), \((544,\pm 15870)\)
Invariants
| Conductor: | $N$ | = | \( 412620 \) | = | $2^{2} \cdot 3 \cdot 5 \cdot 13 \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $-324234125523360000$ | = | $-1 \cdot 2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 13^{2} \cdot 23^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{14647977776}{8555625} \) | = | $2^{4} \cdot 3^{-4} \cdot 5^{-4} \cdot 13^{-2} \cdot 971^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0494393470920232892782857187$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.019594118754151570930087888489$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9914250933349535$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6940965068308933$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.8093134794444246505765552823$ |
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| Real period: | $\Omega$ | ≈ | $0.17986483628644874098634924966$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 192 $ = $ 3\cdot2^{2}\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $15.620729893014546198086196256 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 15.620729893 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.179865 \cdot 1.809313 \cdot 192}{2^2} \\ & \approx 15.620729893\end{aligned}$$
Modular invariants
Modular form 412620.2.a.bo
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4730880 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $13$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $23$ | $2$ | $I_0^{*}$ | 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 | 4.6.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 11960 = 2^{3} \cdot 5 \cdot 13 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1559 & 0 \\ 0 & 11959 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 10 & 27 \end{array}\right),\left(\begin{array}{rr} 11953 & 8 \\ 11952 & 9 \end{array}\right),\left(\begin{array}{rr} 3681 & 8326 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 8971 & 1564 \\ 11247 & 3129 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 5981 & 3128 \\ 6762 & 6257 \end{array}\right),\left(\begin{array}{rr} 7177 & 1564 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[11960])$ is a degree-$107549782179840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/11960\Z)$.
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 | $2$ | \( 529 = 23^{2} \) |
| $3$ | split multiplicative | $4$ | \( 137540 = 2^{2} \cdot 5 \cdot 13 \cdot 23^{2} \) |
| $5$ | split multiplicative | $6$ | \( 82524 = 2^{2} \cdot 3 \cdot 13 \cdot 23^{2} \) |
| $13$ | nonsplit multiplicative | $14$ | \( 31740 = 2^{2} \cdot 3 \cdot 5 \cdot 23^{2} \) |
| $23$ | additive | $266$ | \( 780 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 412620.bo
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 780.c2, 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.