Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+475225x-1154084875\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+475225xz^2-1154084875z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+615890925x-53854222295250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(43045/16, 8875135/64)$ | $6.7297610214825074934610441701$ | $\infty$ |
| $(3595/4, -3595/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 8450 \) | = | $2 \cdot 5^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-582452128062025000000$ | = | $-1 \cdot 2^{6} \cdot 5^{8} \cdot 13^{12} $ |
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| j-invariant: | $j$ | = | \( \frac{157376536199}{7722894400} \) | = | $2^{-6} \cdot 5^{-2} \cdot 13^{-6} \cdot 5399^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6642166277054547956255758928$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.57702299275763624029845250540$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0187728994158647$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.111160599004555$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.7297610214825074934610441701$ |
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| Real period: | $\Omega$ | ≈ | $0.078229306593537666106308411746$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.2117163060063544199775246927 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.211716306 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.078229 \cdot 6.729761 \cdot 32}{2^2} \\ & \approx 4.211716306\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 580608 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $13$ | $4$ | $I_{6}^{*}$ | additive | 1 | 2 | 12 | 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 | 4.12.0.6 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 13 & 24 \\ 852 & 253 \end{array}\right),\left(\begin{array}{rr} 1537 & 24 \\ 1536 & 25 \end{array}\right),\left(\begin{array}{rr} 479 & 1554 \\ 1068 & 1487 \end{array}\right),\left(\begin{array}{rr} 781 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 603 & 1550 \\ 1036 & 1459 \end{array}\right),\left(\begin{array}{rr} 391 & 24 \\ 195 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 16 \\ 314 & 335 \end{array}\right),\left(\begin{array}{rr} 521 & 24 \\ 1300 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$2415329280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $4$ | \( 4225 = 5^{2} \cdot 13^{2} \) |
| $3$ | good | $2$ | \( 4225 = 5^{2} \cdot 13^{2} \) |
| $5$ | additive | $18$ | \( 338 = 2 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 50 = 2 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 8450.i
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 130.a4, its twist by $65$.
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 |
| $2$ | \(\Q(\sqrt{-195}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.16900.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{195})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.5005040625.6 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.4569760000.4 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.23134410000.2 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.1951615085175320647658814825795000000000000.3 | \(\Z/18\Z\) | 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 | ord | add | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 1 | - | 1 | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | 1 | - | 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.