Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-35232157875x-2545405502660750\)
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(homogenize, simplify) |
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\(y^2z=x^3-35232157875xz^2-2545405502660750z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-35232157875x-2545405502660750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-108370, 0)$ | $0$ | $2$ |
Integral points
\( \left(-108370, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 478800 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $278160318922752000000$ | = | $2^{30} \cdot 3^{8} \cdot 5^{6} \cdot 7 \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{103665426767620308239307625}{5961940992} \) | = | $2^{-18} \cdot 3^{-2} \cdot 5^{3} \cdot 7^{-1} \cdot 17^{3} \cdot 19^{-2} \cdot 5526613^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.3094055171109823485233277501$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.2622332359999320061080933436$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0583334848509944$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.458370763689036$ | |||
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: | $\Omega$ | ≈ | $0.011019572430946455062187929697$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.1736368601125790579101237528 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $36$ = $6^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.173636860 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{36 \cdot 0.011020 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 3.173636860\end{aligned}$$
Modular invariants
Modular form 478800.2.a.pj
For more coefficients, see the Downloads section to the right.
| Modular degree: | 573308928 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{22}^{*}$ | additive | -1 | 4 | 30 | 18 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $19$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 | 2.3.0.1 |
| $3$ | 3B | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 23940 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 19 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 14354 & 19125 \\ 7925 & 6794 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 11699 & 9540 \\ 9250 & 11059 \end{array}\right),\left(\begin{array}{rr} 14363 & 0 \\ 0 & 23939 \end{array}\right),\left(\begin{array}{rr} 2551 & 14400 \\ 13670 & 19411 \end{array}\right),\left(\begin{array}{rr} 3436 & 9585 \\ 9875 & 18166 \end{array}\right),\left(\begin{array}{rr} 23905 & 36 \\ 23904 & 37 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 3240 & 6139 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 14 & 253 \end{array}\right)$.
The torsion field $K:=\Q(E[23940])$ is a degree-$51468809011200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/23940\Z)$.
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 | $2$ | \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \) |
| $3$ | additive | $8$ | \( 53200 = 2^{4} \cdot 5^{2} \cdot 7 \cdot 19 \) |
| $5$ | additive | $14$ | \( 19152 = 2^{4} \cdot 3^{2} \cdot 7 \cdot 19 \) |
| $7$ | split multiplicative | $8$ | \( 68400 = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 25200 = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 478800.pj
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 798.d1, its twist by $60$.
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 $0$.