Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-246033x-4422438\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-246033xz^2-4422438z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-19928700x-3283743375\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-139, 5203)$ | $3.4429947255035826266499435384$ | $\infty$ |
| $(-18, 0)$ | $0$ | $2$ |
Integral points
\((-139,\pm 5203)\), \( \left(-18, 0\right) \), \((2007,\pm 87075)\)
Invariants
| Conductor: | $N$ | = | \( 471900 \) | = | $2^{2} \cdot 3 \cdot 5^{2} \cdot 11^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $944385952331250000$ | = | $2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 11^{6} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{3718856704}{2132325} \) | = | $2^{14} \cdot 3^{-8} \cdot 5^{-2} \cdot 13^{-1} \cdot 61^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1387852414637112484976446976$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.095930411339172647306117465147$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0723611999657963$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.7393843067181622$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.4429947255035826266499435384$ |
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| Real period: | $\Omega$ | ≈ | $0.23262631206513055114216181276$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 3\cdot2\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.8055869927215694796587224447 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.805586993 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.232626 \cdot 3.442995 \cdot 24}{2^2} \\ & \approx 4.805586993\end{aligned}$$
Modular invariants
Modular form 471900.2.a.p
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4976640 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $5$ | $2$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $11$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 | 4.6.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5720 = 2^{3} \cdot 5 \cdot 11 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3257 & 4686 \\ 1034 & 1035 \end{array}\right),\left(\begin{array}{rr} 3321 & 1034 \\ 3542 & 4685 \end{array}\right),\left(\begin{array}{rr} 5446 & 3641 \\ 1463 & 5204 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 5713 & 8 \\ 5712 & 9 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 28 & 75 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 3639 & 0 \\ 0 & 5719 \end{array}\right),\left(\begin{array}{rr} 2861 & 528 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[5720])$ is a degree-$5313724416000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5720\Z)$.
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 | $2$ | \( 39325 = 5^{2} \cdot 11^{2} \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 157300 = 2^{2} \cdot 5^{2} \cdot 11^{2} \cdot 13 \) |
| $5$ | additive | $18$ | \( 18876 = 2^{2} \cdot 3 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $62$ | \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 36300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 471900.p
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 780.c1, its twist by $-55$.
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.