Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-34740300x-93157702000\)
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(homogenize, simplify) |
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\(y^2z=x^3-34740300xz^2-93157702000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-34740300x-93157702000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{476960}{49}, \frac{240333500}{343}\right) \) | $10.277256583871412166993017960$ | $\infty$ |
| \( \left(6940, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([3338720:240333500:343]\) | $10.277256583871412166993017960$ | $\infty$ |
| \([6940:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{476960}{49}, \frac{240333500}{343}\right) \) | $10.277256583871412166993017960$ | $\infty$ |
| \( \left(6940, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(6940, 0\right) \)
\([6940:0:1]\)
\( \left(6940, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 417600 \) | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 29$ |
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| Minimal Discriminant: | $\Delta$ | = | $-1065679751599718400000000$ | = | $-1 \cdot 2^{21} \cdot 3^{7} \cdot 5^{8} \cdot 29^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{1552876541267401}{356893992600} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-1} \cdot 5^{-2} \cdot 7^{3} \cdot 29^{-6} \cdot 71^{3} \cdot 233^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.3301395802386838254054364886$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.93639370884766082828158602134$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9786207769286249$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.9479897014355405$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $10.277256583871412166993017960$ |
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| Real period: | $\Omega$ | ≈ | $0.030717912449640824482602476102$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.0511338874537085128418749127 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.051133887 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.030718 \cdot 10.277257 \cdot 64}{2^2} \\ & \approx 5.051133887\end{aligned}$$
Modular invariants
Modular form 417600.2.a.y
For more coefficients, see the Downloads section to the right.
| Modular degree: | 63700992 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{11}^{*}$ | additive | -1 | 6 | 21 | 3 |
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $29$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3480 = 2^{3} \cdot 3 \cdot 5 \cdot 29 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 2783 & 3468 \\ 2778 & 3407 \end{array}\right),\left(\begin{array}{rr} 2890 & 3477 \\ 1187 & 8 \end{array}\right),\left(\begin{array}{rr} 3470 & 3477 \\ 1767 & 8 \end{array}\right),\left(\begin{array}{rr} 1005 & 1592 \\ 1042 & 1603 \end{array}\right),\left(\begin{array}{rr} 3469 & 12 \\ 3468 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 2641 & 12 \\ 1926 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 3430 & 3471 \end{array}\right)$.
The torsion field $K:=\Q(E[3480])$ is a degree-$251441971200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3480\Z)$.
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 | $4$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $3$ | additive | $8$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
| $5$ | additive | $18$ | \( 16704 = 2^{6} \cdot 3^{2} \cdot 29 \) |
| $29$ | nonsplit multiplicative | $30$ | \( 14400 = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 417600y
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 870c4, its twist by $120$.
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.