Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2+72067x-30335763\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z+72067xz^2-30335763z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+5837400x-22097259000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(237, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([237:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(2130, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(237, 0\right) \)
\([237:0:1]\)
\( \left(237, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 436800 \) | = | $2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13$ |
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| Minimal Discriminant: | $\Delta$ | = | $-420876591408000000$ | = | $-1 \cdot 2^{10} \cdot 3^{3} \cdot 5^{6} \cdot 7^{8} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{2587063175168}{26304786963} \) | = | $2^{11} \cdot 3^{-3} \cdot 7^{-8} \cdot 13^{-2} \cdot 23^{3} \cdot 47^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0628605023857101499829646061$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.68051889570203887150155817161$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0143277708927247$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6941853109794027$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
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: | $\Omega$ | ≈ | $0.14722959733889391580919392274$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot1\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.3556735574223026529471027638 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.355673557 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.147230 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 2.355673557\end{aligned}$$
Modular invariants
Modular form 436800.2.a.fa
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4718592 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | 1 | 6 | 10 | 0 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $13$ | $2$ | $I_{2}$ | split 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.12.0.5 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 21840 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 16941 & 13120 \\ 11690 & 12251 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21825 & 16 \\ 21824 & 17 \end{array}\right),\left(\begin{array}{rr} 4381 & 13120 \\ 3440 & 8421 \end{array}\right),\left(\begin{array}{rr} 21839 & 8720 \\ 16380 & 16379 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 21742 & 21827 \end{array}\right),\left(\begin{array}{rr} 18721 & 13120 \\ 14360 & 17601 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 21836 & 21837 \end{array}\right),\left(\begin{array}{rr} 13103 & 0 \\ 0 & 21839 \end{array}\right),\left(\begin{array}{rr} 14576 & 5 \\ 21795 & 21826 \end{array}\right)$.
The torsion field $K:=\Q(E[21840])$ is a degree-$155817722511360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/21840\Z)$.
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 | $2$ | \( 75 = 3 \cdot 5^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 145600 = 2^{6} \cdot 5^{2} \cdot 7 \cdot 13 \) |
| $5$ | additive | $14$ | \( 17472 = 2^{6} \cdot 3 \cdot 7 \cdot 13 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 62400 = 2^{6} \cdot 3 \cdot 5^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 33600 = 2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 436800fa
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 2184j1, its twist by $40$.
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 $0$.