Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-1274033x+553877937\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-1274033xz^2+553877937z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-103196700x+403467426000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(697, 2000)$ | $1.6651681044808013392713239243$ | $\infty$ |
| $(641, 432)$ | $3.1286189111570303367933776201$ | $\infty$ |
| $(647, 0)$ | $0$ | $2$ |
| $(657, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1303, 0\right) \), \((-568,\pm 33075)\), \((-553,\pm 33000)\), \((257,\pm 15600)\), \((608,\pm 1911)\), \((641,\pm 432)\), \( \left(647, 0\right) \), \( \left(657, 0\right) \), \((697,\pm 2000)\), \((1147,\pm 24500)\), \((3793,\pm 224224)\), \((3897,\pm 234000)\), \((19405147,\pm 85482141500)\)
Invariants
| Conductor: | $N$ | = | \( 436800 \) | = | $2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $23372294400000000$ | = | $2^{14} \cdot 3^{2} \cdot 5^{8} \cdot 7^{4} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{893359210685776}{91298025} \) | = | $2^{4} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-4} \cdot 13^{-2} \cdot 37^{3} \cdot 1033^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1739856239606380088429497364$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.56059495709031829388913259475$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9231142529338714$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.141506307102171$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.8339633214384742420807038533$ |
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| Real period: | $\Omega$ | ≈ | $0.36404330689040034395435860243$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $14.078175943386923542035799582 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.078175943 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.364043 \cdot 4.833963 \cdot 128}{4^2} \\ & \approx 14.078175943\end{aligned}$$
Modular invariants
Modular form 436800.2.a.eu
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6291456 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | 1 | 6 | 14 | 0 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $13$ | $2$ | $I_{2}$ | 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 |
|---|---|---|
| $2$ | 2Cs | 4.12.0.3 |
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 $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 10913 & 8 \\ 10912 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4361 & 10918 \\ 4386 & 5 \end{array}\right),\left(\begin{array}{rr} 3647 & 2 \\ 10902 & 10915 \end{array}\right),\left(\begin{array}{rr} 5 & 8192 \\ 2 & 2731 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 10916 & 10917 \end{array}\right),\left(\begin{array}{rr} 7567 & 6 \\ 7554 & 10915 \end{array}\right),\left(\begin{array}{rr} 7801 & 8 \\ 9364 & 33 \end{array}\right),\left(\begin{array}{rr} 5455 & 8184 \\ 8216 & 5491 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$9738607656960$ 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 | $2$ | \( 25 = 5^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 145600 = 2^{6} \cdot 5^{2} \cdot 7 \cdot 13 \) |
| $5$ | additive | $18$ | \( 17472 = 2^{6} \cdot 3 \cdot 7 \cdot 13 \) |
| $7$ | 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 and 4.
Its isogeny class 436800eu
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 10920n2, 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.