Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-231344607x-1354359539806\)
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(homogenize, simplify) |
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\(y^2z=x^3-231344607xz^2-1354359539806z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-231344607x-1354359539806\)
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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 |
|---|---|---|
| \( \left(-8762, 0\right) \) | $0$ | $2$ |
| \( \left(17563, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-8762:0:1]\) | $0$ | $2$ |
| \([17563:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-8762, 0\right) \) | $0$ | $2$ |
| \( \left(17563, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-8801, 0\right) \), \( \left(-8762, 0\right) \), \( \left(17563, 0\right) \)
\([-8801:0:1]\), \([-8762:0:1]\), \([17563:0:1]\)
\( \left(-8801, 0\right) \), \( \left(-8762, 0\right) \), \( \left(17563, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 60840 \) | = | $2^{3} \cdot 3^{2} \cdot 5 \cdot 13^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $11722184762675905440000$ | = | $2^{8} \cdot 3^{12} \cdot 5^{4} \cdot 13^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{1520107298839022416}{13013105625} \) | = | $2^{4} \cdot 3^{-6} \cdot 5^{-4} \cdot 13^{-4} \cdot 181^{3} \cdot 2521^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.4027784886334226026111491452$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1088995451953025159419613917$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0142518435034014$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.299186875357685$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
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: | $\Omega$ | ≈ | $0.038711048349334654140827587159$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 256 $ = $ 2^{2}\cdot2^{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)$ | ≈ | $0.61937677358935446625324139455 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.619376774 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.038711 \cdot 1.000000 \cdot 256}{4^2} \\ & \approx 0.619376774\end{aligned}$$
Modular invariants
Modular form 60840.2.a.bm
For more coefficients, see the Downloads section to the right.
| Modular degree: | 10321920 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{1}^{*}$ | additive | -1 | 3 | 8 | 0 |
| $3$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $13$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 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 |
|---|---|---|---|
| $2$ | 2Cs | 8.48.0.133 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 312 = 2^{3} \cdot 3 \cdot 13 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 308 & 309 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 97 & 306 \\ 6 & 5 \end{array}\right),\left(\begin{array}{rr} 161 & 240 \\ 208 & 125 \end{array}\right),\left(\begin{array}{rr} 5 & 80 \\ 2 & 235 \end{array}\right),\left(\begin{array}{rr} 305 & 8 \\ 304 & 9 \end{array}\right),\left(\begin{array}{rr} 167 & 304 \\ 44 & 279 \end{array}\right)$.
The torsion field $K:=\Q(E[312])$ is a degree-$10063872$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/312\Z)$.
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 | $2$ | \( 1521 = 3^{2} \cdot 13^{2} \) |
| $3$ | additive | $2$ | \( 6760 = 2^{3} \cdot 5 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 12168 = 2^{3} \cdot 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 60840br
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 1560j2, its twist by $-39$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | \(\Q(i, \sqrt{3}, \sqrt{13})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.53301680640000.21 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | \(\Q(i, \sqrt{6}, \sqrt{26})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | \(\Q(\sqrt{-2}, \sqrt{3}, \sqrt{-13})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | \(\Q(i, \sqrt{2}, \sqrt{3}, \sqrt{13})\) | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 13 |
|---|---|---|---|---|
| Reduction type | add | add | split | add |
| $\lambda$-invariant(s) | - | - | 1 | - |
| $\mu$-invariant(s) | - | - | 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 $0$.