Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-64304728033x+6276448294455937\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-64304728033xz^2+6276448294455937z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5208682970700x+4575515180609466000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(145197, 800800)$ | $1.1675016209609289318201444585$ | $\infty$ |
| $(146432, 16575)$ | $3.0514464186336643290917010897$ | $\infty$ |
| $(146407, 0)$ | $0$ | $2$ |
Integral points
\((-146003,\pm 112039200)\), \((111597,\pm 22136800)\), \((145197,\pm 800800)\), \( \left(146407, 0\right) \), \((146432,\pm 16575)\), \((146731,\pm 214812)\), \((156208,\pm 6567561)\)
Invariants
| Conductor: | $N$ | = | \( 436800 \) | = | $2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $202513585397760000000$ | = | $2^{19} \cdot 3^{8} \cdot 5^{7} \cdot 7^{3} \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{7179471593960193209684686321}{49441793310} \) | = | $2^{-1} \cdot 3^{-8} \cdot 5^{-1} \cdot 7^{-3} \cdot 11^{6} \cdot 13^{-3} \cdot 107^{3} \cdot 109^{3} \cdot 1367^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.4322214777837868468126741751$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.5877817507268186953864463263$ |
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| $abc$ quality: | $Q$ | ≈ | $1.087781017100558$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.6430097371757135$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.2827396905565842833235477678$ |
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| Real period: | $\Omega$ | ≈ | $0.059626937257651927052522477969$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 288 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot3\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $14.093259376465523036205259170 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.093259376 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.059627 \cdot 3.282740 \cdot 288}{2^2} \\ & \approx 14.093259376\end{aligned}$$
Modular invariants
Modular form 436800.2.a.hy
For more coefficients, see the Downloads section to the right.
| Modular degree: | 509607936 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 6 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{9}^{*}$ | additive | -1 | 6 | 19 | 1 |
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $13$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 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 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 9614 & 1691 \end{array}\right),\left(\begin{array}{rr} 8186 & 10917 \\ 8145 & 10886 \end{array}\right),\left(\begin{array}{rr} 2168 & 10899 \\ 285 & 374 \end{array}\right),\left(\begin{array}{rr} 10897 & 24 \\ 10896 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1821 & 3644 \\ 20 & 5541 \end{array}\right),\left(\begin{array}{rr} 1696 & 21 \\ 3915 & 10546 \end{array}\right),\left(\begin{array}{rr} 4696 & 3 \\ 1101 & 10834 \end{array}\right),\left(\begin{array}{rr} 2274 & 431 \\ 5005 & 3184 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$4869303828480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 | $4$ | \( 2275 = 5^{2} \cdot 7 \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
| $5$ | additive | $18$ | \( 17472 = 2^{6} \cdot 3 \cdot 7 \cdot 13 \) |
| $7$ | split 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, 3, 4, 6 and 12.
Its isogeny class 436800.hy
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 2730.bd1, its twist by $-40$.
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.