Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+71896232x+205235610688\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+71896232xz^2+205235610688z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+93177515997x+9574074989515998\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(11447755550688/8692392289, 444981202648952488408/810417810280337)$ | $30.786326094257829564934723553$ | $\infty$ |
| $(-2608, 1304)$ | $0$ | $2$ |
Integral points
\( \left(-2608, 1304\right) \)
Invariants
| Conductor: | $N$ | = | \( 479370 \) | = | $2 \cdot 3 \cdot 5 \cdot 19 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $-41975980763686201500000000$ | = | $-1 \cdot 2^{8} \cdot 3 \cdot 5^{9} \cdot 19^{6} \cdot 29^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{69096190760262356111}{70568821500000000} \) | = | $2^{-8} \cdot 3^{-1} \cdot 5^{-9} \cdot 19^{-6} \cdot 29^{3} \cdot 141499^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.6034337457988848447224340205$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.9197858308056478311307980043$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0444928492259786$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.037053341257333$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $30.786326094257829564934723553$ |
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| Real period: | $\Omega$ | ≈ | $0.042455731553617277990101980211$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot1\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.6141119923598662165427660495 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.614111992 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.042456 \cdot 30.786326 \cdot 8}{2^2} \\ & \approx 2.614111992\end{aligned}$$
Modular invariants
Modular form 479370.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 217728000 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{9}$ | nonsplit multiplicative | 1 | 1 | 9 | 9 |
| $19$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $29$ | $2$ | $I_0^{*}$ | 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 | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 33060 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \cdot 29 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 9890 & 21663 \\ 14413 & 1132 \end{array}\right),\left(\begin{array}{rr} 2379 & 24418 \\ 16646 & 28421 \end{array}\right),\left(\begin{array}{rr} 18239 & 0 \\ 0 & 33059 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 33010 & 33051 \end{array}\right),\left(\begin{array}{rr} 31930 & 21663 \\ 29841 & 1132 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 12181 & 20532 \\ 17226 & 24013 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 33049 & 12 \\ 33048 & 13 \end{array}\right)$.
The torsion field $K:=\Q(E[33060])$ is a degree-$1934845968384000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/33060\Z)$.
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 | $4$ | \( 12615 = 3 \cdot 5 \cdot 29^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 1682 = 2 \cdot 29^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 95874 = 2 \cdot 3 \cdot 19 \cdot 29^{2} \) |
| $19$ | nonsplit multiplicative | $20$ | \( 25230 = 2 \cdot 3 \cdot 5 \cdot 29^{2} \) |
| $29$ | additive | $422$ | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 479370.h
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 570.k4, its twist by $29$.
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.