Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-16970992908417x+26919090475475527741\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-16970992908417xz^2+26919090475475527741z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-271535886534675x+1722821518894547240750\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 418950 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $-218072650466268168722013392732160000000$ | = | $-1 \cdot 2^{81} \cdot 3^{11} \cdot 5^{7} \cdot 7^{3} \cdot 19 $ |
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| j-invariant: | $j$ | = | \( -\frac{138357846491853121383730987168838623}{55816105091607428996184145920} \) | = | $-1 \cdot 2^{-81} \cdot 3^{-5} \cdot 5^{-1} \cdot 19^{-1} \cdot 322351^{3} \cdot 1604497^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $6.3197920553680332683928851994$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $4.4792894175530999091185447285$ |
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| $abc$ quality: | $Q$ | ≈ | $1.082511858129502$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.956573745607039$ | |||
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: | $\Omega$ | ≈ | $0.0055118339596330270673713378044$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2^{2}\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.088189343354128433077941404870 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.088189343 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.005512 \cdot 1.000000 \cdot 16}{1^2} \\ & \approx 0.088189343\end{aligned}$$
Modular invariants
Modular form 418950.2.a.m
For more coefficients, see the Downloads section to the right.
| Modular degree: | 26266705920 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{81}$ | nonsplit multiplicative | 1 | 1 | 81 | 81 |
| $3$ | $4$ | $I_{5}^{*}$ | additive | -1 | 2 | 11 | 5 |
| $5$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
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 $2$, genus $0$, and generators
$\left(\begin{array}{rr} 9577 & 2 \\ 9577 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 15959 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3991 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15959 & 2 \\ 15958 & 3 \end{array}\right),\left(\begin{array}{rr} 13681 & 2 \\ 13681 & 3 \end{array}\right),\left(\begin{array}{rr} 5321 & 2 \\ 5321 & 3 \end{array}\right),\left(\begin{array}{rr} 4201 & 2 \\ 4201 & 3 \end{array}\right),\left(\begin{array}{rr} 7981 & 2 \\ 7981 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[15960])$ is a degree-$4392005035622400$ 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 | $4$ | \( 29925 = 3^{2} \cdot 5^{2} \cdot 7 \cdot 19 \) |
| $3$ | 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 | $20$ | \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 22050 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 418950.m consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 27930.bi1, its twist by $-15$.
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 $0$.