Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-889632x-322674944\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-889632xz^2-322674944z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-72060219x-235446214806\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(77034/25, 20226602/125)$ | $8.3236044047770941437293050216$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 25168 \) | = | $2^{4} \cdot 11^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-48298025025536$ | = | $-1 \cdot 2^{21} \cdot 11^{6} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{10730978619193}{6656} \) | = | $-1 \cdot 2^{-9} \cdot 7^{3} \cdot 13^{-1} \cdot 23^{3} \cdot 137^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9464812430925072372214144211$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.054386426133376655773210510660$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0219339430875438$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.201577556279655$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $8.3236044047770941437293050216$ |
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| Real period: | $\Omega$ | ≈ | $0.077726104679758900180839094598$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.5878453891144267678598410186 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.587845389 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.077726 \cdot 8.323604 \cdot 4}{1^2} \\ & \approx 2.587845389\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 155520 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{13}^{*}$ | additive | -1 | 4 | 21 | 9 |
| $11$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $3$ | 3B | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10296 = 2^{3} \cdot 3^{2} \cdot 11 \cdot 13 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9359 & 0 \\ 0 & 10295 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 4192 & 1881 \\ 3663 & 1948 \end{array}\right),\left(\begin{array}{rr} 3266 & 8415 \\ 4455 & 1880 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 7723 & 8910 \\ 0 & 8867 \end{array}\right),\left(\begin{array}{rr} 1882 & 1881 \\ 3267 & 8416 \end{array}\right),\left(\begin{array}{rr} 10279 & 18 \\ 10278 & 19 \end{array}\right)$.
The torsion field $K:=\Q(E[10296])$ is a degree-$14347055923200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10296\Z)$.
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 | $4$ | \( 1573 = 11^{2} \cdot 13 \) |
| $11$ | additive | $62$ | \( 208 = 2^{4} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 1936 = 2^{4} \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 25168bb
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 26a2, its twist by $44$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-33}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.104.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.1773613423296.2 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.47887562428992.1 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.6219113472.5 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.2.3862100346548736317358655106853854735499264.1 | \(\Z/6\Z\) | not in database |
| $18$ | 18.0.76017721121118776934570408468204422758832013312.1 | \(\Z/18\Z\) | 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 | add | ord | ord | ord | add | nonsplit | ord | ord | ss | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | - | 1 | 1 | 1 | - | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1,1 | 1 | 1 |
| $\mu$-invariant(s) | - | 2 | 0 | 0 | - | 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.