Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-108269179633x+13712108457168863\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-108269179633xz^2+13712108457168863z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-8769803550300x+9996153374686752000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(189973, 0)$ | $0$ | $2$ |
Integral points
\( \left(189973, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 405600 \) | = | $2^{5} \cdot 3 \cdot 5^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $927681605150040000000000$ | = | $2^{12} \cdot 3^{7} \cdot 5^{10} \cdot 13^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{454357982636417669333824}{3003024375} \) | = | $2^{6} \cdot 3^{-7} \cdot 5^{-4} \cdot 13^{-3} \cdot 47^{3} \cdot 408923^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.6579064944219353606519552732$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.8775656789141714959075997643$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1653240287550397$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.80217047125869$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.043249438421271169149060271341$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 112 $ = $ 2\cdot7\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.2109842757955927361736875975 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.210984276 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.043249 \cdot 1.000000 \cdot 112}{2^2} \\ & \approx 1.210984276\end{aligned}$$
Modular invariants
Modular form 405600.2.a.fh
For more coefficients, see the Downloads section to the right.
| Modular degree: | 867041280 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{3}^{*}$ | additive | 1 | 5 | 12 | 0 |
| $3$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $5$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
| $13$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 203 & 198 \\ 1370 & 587 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 1553 & 8 \\ 1552 & 9 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1554 & 1555 \end{array}\right),\left(\begin{array}{rr} 356 & 1559 \\ 97 & 1554 \end{array}\right),\left(\begin{array}{rr} 976 & 203 \\ 979 & 1008 \end{array}\right),\left(\begin{array}{rr} 1048 & 3 \\ 1045 & 2 \end{array}\right),\left(\begin{array}{rr} 623 & 1552 \\ 932 & 1527 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$19322634240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $2$ | \( 12675 = 3 \cdot 5^{2} \cdot 13^{2} \) |
| $3$ | split multiplicative | $4$ | \( 135200 = 2^{5} \cdot 5^{2} \cdot 13^{2} \) |
| $5$ | additive | $18$ | \( 16224 = 2^{5} \cdot 3 \cdot 13^{2} \) |
| $7$ | good | $2$ | \( 135200 = 2^{5} \cdot 5^{2} \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 405600.fh
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 6240.c1, its twist by $65$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.