Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-3510x+115425\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-3510xz^2+115425z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-56163x+7331038\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{285}{4}, \frac{285}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-570:285:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-286, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 1305 \) | = | $3^{2} \cdot 5 \cdot 29$ |
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| Minimal Discriminant: | $\Delta$ | = | $-2900294150625$ | = | $-1 \cdot 3^{8} \cdot 5^{4} \cdot 29^{4} $ |
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| j-invariant: | $j$ | = | \( -\frac{6561258219361}{3978455625} \) | = | $-1 \cdot 3^{-2} \cdot 5^{-4} \cdot 29^{-4} \cdot 97^{3} \cdot 193^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0934255766907937330569549530$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.54411943235673888735933233454$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9529824634176626$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.132540931664702$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.74405718460293353098531431139$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.4881143692058670619706286228 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.488114369 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.744057 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 1.488114369\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2560 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $29$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.24.0.75 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6960 = 2^{4} \cdot 3 \cdot 5 \cdot 29 \), index $192$, genus $3$, and generators
$\left(\begin{array}{rr} 4177 & 16 \\ 2788 & 65 \end{array}\right),\left(\begin{array}{rr} 2641 & 16 \\ 3604 & 65 \end{array}\right),\left(\begin{array}{rr} 5 & 16 \\ 64 & 205 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6945 & 16 \\ 6944 & 17 \end{array}\right),\left(\begin{array}{rr} 11 & 4366 \\ 1070 & 3771 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 4 & 49 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 6945 & 6794 \\ 3926 & 3853 \end{array}\right),\left(\begin{array}{rr} 2176 & 4801 \\ 4785 & 2176 \end{array}\right)$.
The torsion field $K:=\Q(E[6960])$ is a degree-$2011535769600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6960\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 9 = 3^{2} \) |
| $3$ | additive | $8$ | \( 145 = 5 \cdot 29 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 261 = 3^{2} \cdot 29 \) |
| $29$ | nonsplit multiplicative | $30$ | \( 45 = 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 1305c
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 435d4, its twist by $-3$.
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 |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/4\Z\) | 2.0.3.1-63075.1-c1 |
| $4$ | \(\Q(\zeta_{12})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | \(\Q(\zeta_{24})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.2346588610560000.26 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.572897610000.2 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.78307942066875.2 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | 29 |
|---|---|---|---|---|
| Reduction type | ord | add | nonsplit | nonsplit |
| $\lambda$-invariant(s) | 2 | - | 0 | 0 |
| $\mu$-invariant(s) | 1 | - | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.