Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+38162808x+937543761216\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+38162808xz^2+937543761216z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+610604925x+60003411322750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(179664, 76114968)$ | $3.0754542470450509651605090006$ | $\infty$ |
| $(-8496, 4248)$ | $0$ | $2$ |
Integral points
\( \left(-8496, 4248\right) \), \( \left(179664, 76114968\right) \), \( \left(179664, -76294632\right) \)
Invariants
| Conductor: | $N$ | = | \( 418950 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $-383287815588072346560000000$ | = | $-1 \cdot 2^{12} \cdot 3^{13} \cdot 5^{7} \cdot 7^{8} \cdot 19^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{4586790226340951}{286015269335040} \) | = | $2^{-12} \cdot 3^{-7} \cdot 5^{-1} \cdot 7^{-2} \cdot 19^{-4} \cdot 166151^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.7805191345923279773271443860$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4535389595135662917764657292$ |
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| $abc$ quality: | $Q$ | ≈ | $1.001557491152603$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.303479050838512$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.0754542470450509651605090006$ |
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| Real period: | $\Omega$ | ≈ | $0.040759343079321111973408650798$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2\cdot2^{2}\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.0113118329620614220932452872 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.011311833 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.040759 \cdot 3.075454 \cdot 128}{2^2} \\ & \approx 4.011311833\end{aligned}$$
Modular invariants
Modular form 418950.2.a.x
For more coefficients, see the Downloads section to the right.
| Modular degree: | 148635648 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $3$ | $4$ | $I_{7}^{*}$ | additive | -1 | 2 | 13 | 7 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $19$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 15960 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 15954 & 15955 \end{array}\right),\left(\begin{array}{rr} 2003 & 13398 \\ 11690 & 15107 \end{array}\right),\left(\begin{array}{rr} 2279 & 0 \\ 0 & 15959 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 15953 & 8 \\ 15952 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 5312 & 4557 \\ 7595 & 6838 \end{array}\right),\left(\begin{array}{rr} 14132 & 6839 \\ 889 & 2274 \end{array}\right),\left(\begin{array}{rr} 4201 & 9128 \\ 5404 & 4593 \end{array}\right),\left(\begin{array}{rr} 5993 & 1428 \\ 3710 & 7127 \end{array}\right)$.
The torsion field $K:=\Q(E[15960])$ is a degree-$183000209817600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/15960\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 11025 = 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
| $3$ | additive | $8$ | \( 23275 = 5^{2} \cdot 7^{2} \cdot 19 \) |
| $5$ | additive | $18$ | \( 16758 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 19 \) |
| $7$ | additive | $32$ | \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 22050 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 418950.x
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 3990.l4, its twist by $105$.
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.