Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-556542x-178473884\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-556542xz^2-178473884z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-8904675x-11431233250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1404, 41798)$ | $1.2470373815073425281910700935$ | $\infty$ |
| $(3491/4, -3491/8)$ | $0$ | $2$ |
Integral points
\( \left(1404, 41798\right) \), \( \left(1404, -43202\right) \), \( \left(4124, 258038\right) \), \( \left(4124, -262162\right) \)
Invariants
| Conductor: | $N$ | = | \( 7650 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-2749419968906250000$ | = | $-1 \cdot 2^{4} \cdot 3^{6} \cdot 5^{10} \cdot 17^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{1673672305534489}{241375690000} \) | = | $-1 \cdot 2^{-4} \cdot 5^{-4} \cdot 13^{3} \cdot 17^{-6} \cdot 9133^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2687690235390560373740559442$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.91474392298795100437605365913$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9921013018955642$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.761793112899798$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.2470373815073425281910700935$ |
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| Real period: | $\Omega$ | ≈ | $0.086704077744032237286733153093$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2\cdot2\cdot2^{2}\cdot( 2 \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.5949574258222483737460159767 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.594957426 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.086704 \cdot 1.247037 \cdot 96}{2^2} \\ & \approx 2.594957426\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 184320 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
| $17$ | $6$ | $I_{6}$ | split 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.5 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 1021 & 24 \\ 6 & 145 \end{array}\right),\left(\begin{array}{rr} 1341 & 2024 \\ 194 & 1917 \end{array}\right),\left(\begin{array}{rr} 2017 & 24 \\ 2016 & 25 \end{array}\right),\left(\begin{array}{rr} 981 & 10 \\ 92 & 101 \end{array}\right),\left(\begin{array}{rr} 13 & 24 \\ 1332 & 733 \end{array}\right),\left(\begin{array}{rr} 511 & 24 \\ 261 & 145 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 16 \\ 314 & 335 \end{array}\right),\left(\begin{array}{rr} 1203 & 2036 \\ 1984 & 67 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[2040])$ is a degree-$7219445760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2040\Z)$.
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 | $4$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $3$ | additive | $2$ | \( 50 = 2 \cdot 5^{2} \) |
| $5$ | additive | $18$ | \( 306 = 2 \cdot 3^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 7650.g
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 170.a3, its twist by $-15$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.260100.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{15})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.450000.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.1082432160000.15 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.958832640000.74 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.67652010000.1 | \(\Z/12\Z\) | not in database |
| $12$ | 12.0.1822500000000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.51840000000000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.3583672967310433050937500000000.1 | \(\Z/18\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | add | add | ord | ord | ord | split | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | - | - | 1 | 3 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.