Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-1573073833x+25958346058857\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-1573073833xz^2+25958346058857z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-25169181323x+1661308978585542\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(807495, 724358876)$ | $7.5494571851110000899640185780$ | $\infty$ |
Integral points
\( \left(807495, 724358876\right) \), \( \left(807495, -725166372\right) \)
Invariants
| Conductor: | $N$ | = | \( 204490 \) | = | $2 \cdot 5 \cdot 11^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-41957488730147620535412981760$ | = | $-1 \cdot 2^{35} \cdot 5 \cdot 11^{6} \cdot 13^{10} $ |
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| j-invariant: | $j$ | = | \( -\frac{1762712152495281}{171798691840} \) | = | $-1 \cdot 2^{-35} \cdot 3^{3} \cdot 5^{-1} \cdot 13^{2} \cdot 7283^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.2332078668141134419128077016$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.89680243253031422317059637798$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1377369369986685$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.157708082904779$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.5494571851110000899640185780$ |
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| Real period: | $\Omega$ | ≈ | $0.035304181888535244991808063556$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 35 $ = $ ( 5 \cdot 7 )\cdot1\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $9.3284593368003814628205948668 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.328459337 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.035304 \cdot 7.549457 \cdot 35}{1^2} \\ & \approx 9.328459337\end{aligned}$$
Modular invariants
Modular form 204490.2.a.cm
For more coefficients, see the Downloads section to the right.
| Modular degree: | 206388000 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $35$ | $I_{35}$ | split multiplicative | -1 | 1 | 35 | 35 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $11$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $1$ | $II^{*}$ | additive | 1 | 2 | 10 | 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 |
|---|---|---|
| $7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 40040 = 2^{3} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 37313 & 528 \\ 9086 & 28953 \end{array}\right),\left(\begin{array}{rr} 40027 & 14 \\ 40026 & 15 \end{array}\right),\left(\begin{array}{rr} 30031 & 10934 \\ 35497 & 36499 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 16017 & 10934 \\ 17479 & 36499 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 37509 & 32758 \\ 34650 & 7369 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 3639 & 0 \\ 0 & 40039 \end{array}\right),\left(\begin{array}{rr} 20021 & 10934 \\ 5467 & 36499 \end{array}\right)$.
The torsion field $K:=\Q(E[40040])$ is a degree-$5356234211328000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/40040\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 102245 = 5 \cdot 11^{2} \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 20449 = 11^{2} \cdot 13^{2} \) |
| $7$ | good | $2$ | \( 102245 = 5 \cdot 11^{2} \cdot 13^{2} \) |
| $11$ | additive | $62$ | \( 1690 = 2 \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $50$ | \( 1210 = 2 \cdot 5 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 204490.cm
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 1690.h1, its twist by $-143$.
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 |
|---|---|---|---|
| $3$ | 3.1.6760.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1827904000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.6.8305867851281.2 | \(\Z/7\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $18$ | 18.6.2347010319872339035948600286857681391783936000000.1 | \(\Z/14\Z\) | 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
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.