Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-92751x-10841424\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-92751xz^2-10841424z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7512858x-7925936643\)
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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 |
|---|---|---|
| $(-175, 1)$ | $2.7879727542303925887069749731$ | $\infty$ |
| $(23233, 3540879)$ | $9.7142808472442255629154172430$ | $\infty$ |
| $(-176, 0)$ | $0$ | $2$ |
Integral points
\( \left(-176, 0\right) \), \((-175,\pm 1)\), \((353,\pm 529)\), \((23233,\pm 3540879)\), \((69520,\pm 18329784)\)
Invariants
| Conductor: | $N$ | = | \( 444360 \) | = | $2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $1243501467600$ | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 23^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{37256083456}{525} \) | = | $2^{11} \cdot 3^{-1} \cdot 5^{-2} \cdot 7^{-1} \cdot 263^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4609414109033329260987645872$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.33785475724789035577702253586$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9605836908258326$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.5316253939889193$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $26.649547634852924452686110347$ |
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| Real period: | $\Omega$ | ≈ | $0.27357088886753580866588486154$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot1\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $14.581080868768902299800808612 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.581080869 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.273571 \cdot 26.649548 \cdot 8}{2^2} \\ & \approx 14.581080869\end{aligned}$$
Modular invariants
Modular form 444360.2.a.p
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1622016 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 4 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | -1 | 3 | 4 | 0 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $23$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 | 8.12.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 38640 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 23 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 15 & 2 \\ 38542 & 38627 \end{array}\right),\left(\begin{array}{rr} 14008 & 36961 \\ 34799 & 23530 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 26221 & 11776 \\ 27968 & 28245 \end{array}\right),\left(\begin{array}{rr} 31919 & 0 \\ 0 & 38639 \end{array}\right),\left(\begin{array}{rr} 1 & 11776 \\ 28980 & 28981 \end{array}\right),\left(\begin{array}{rr} 19965 & 11776 \\ 33074 & 18171 \end{array}\right),\left(\begin{array}{rr} 18976 & 30245 \\ 9315 & 25186 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 38636 & 38637 \end{array}\right),\left(\begin{array}{rr} 38625 & 16 \\ 38624 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[38640])$ is a degree-$1588427552194560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/38640\Z)$.
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 | $2$ | \( 11109 = 3 \cdot 7 \cdot 23^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 148120 = 2^{3} \cdot 5 \cdot 7 \cdot 23^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 88872 = 2^{3} \cdot 3 \cdot 7 \cdot 23^{2} \) |
| $7$ | split multiplicative | $8$ | \( 63480 = 2^{3} \cdot 3 \cdot 5 \cdot 23^{2} \) |
| $23$ | additive | $266$ | \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 444360p
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 840f1, its twist by $-23$.
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.