Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-369830x+85590172\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-369830xz^2+85590172z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5917275x+5471853750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-701, 350\right) \) | $0$ | $2$ |
| \( \left(319, -160\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-701:350:1]\) | $0$ | $2$ |
| \([319:-160:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-2805, 0\right) \) | $0$ | $2$ |
| \( \left(1275, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-701, 350\right) \), \( \left(319, -160\right) \)
\([-701:350:1]\), \([319:-160:1]\)
\( \left(-2805, 0\right) \), \( \left(1275, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 65025 \) | = | $3^{2} \cdot 5^{2} \cdot 17^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $79458237101390625$ | = | $3^{6} \cdot 5^{6} \cdot 17^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{20346417}{289} \) | = | $3^{3} \cdot 7^{3} \cdot 13^{3} \cdot 17^{-2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0474220613578969374567205291$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.72320971122131613566604906491$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0296285559493121$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.518463726586186$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.34388970654610391073018320614$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.3755588261844156429207328246 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.375558826 \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 0.343890 \cdot 1.000000 \cdot 64}{4^2} \\ & \approx 1.375558826\end{aligned}$$
Modular invariants
Modular form 65025.2.a.u
For more coefficients, see the Downloads section to the right.
| Modular degree: | 589824 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $17$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
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 |
|---|---|---|---|
| $2$ | 2Cs | 16.48.0.20 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8160 = 2^{5} \cdot 3 \cdot 5 \cdot 17 \), index $1536$, genus $53$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2579 & 3240 \\ 7350 & 1259 \end{array}\right),\left(\begin{array}{rr} 2719 & 0 \\ 0 & 8159 \end{array}\right),\left(\begin{array}{rr} 9 & 32 \\ 8104 & 7961 \end{array}\right),\left(\begin{array}{rr} 8129 & 32 \\ 8128 & 33 \end{array}\right),\left(\begin{array}{rr} 29 & 8 \\ 4252 & 1173 \end{array}\right),\left(\begin{array}{rr} 1981 & 4980 \\ 7320 & 1801 \end{array}\right),\left(\begin{array}{rr} 3263 & 0 \\ 0 & 8159 \end{array}\right),\left(\begin{array}{rr} 511 & 2310 \\ 4080 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[8160])$ is a degree-$462044528640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8160\Z)$.
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 |
|---|---|---|---|
| $3$ | additive | $6$ | \( 7225 = 5^{2} \cdot 17^{2} \) |
| $5$ | additive | $14$ | \( 2601 = 3^{2} \cdot 17^{2} \) |
| $17$ | additive | $162$ | \( 225 = 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 65025.u
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 17.a2, its twist by $-255$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-15}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{15}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{255})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1082432160000.1 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.32993039626875.2 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | 17 |
|---|---|---|---|---|
| Reduction type | ord | add | add | add |
| $\lambda$-invariant(s) | ? | - | - | - |
| $\mu$-invariant(s) | ? | - | - | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry ? indicates that the invariants have not yet been computed.
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 $0$.