Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-5064577x+4388596513\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-5064577xz^2+4388596513z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-410230764x+3198056165712\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{293239}{225}, \frac{297712}{3375}\right) \) | $8.2519610819874383350267468160$ | $\infty$ |
| \( \left(1303, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([4398585:297712:3375]\) | $8.2519610819874383350267468160$ | $\infty$ |
| \([1303:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{293164}{25}, \frac{297712}{125}\right) \) | $8.2519610819874383350267468160$ | $\infty$ |
| \( \left(11724, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(1303, 0\right) \)
\([1303:0:1]\)
\( \left(1303, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 30912 \) | = | $2^{6} \cdot 3 \cdot 7 \cdot 23$ |
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| Minimal Discriminant: | $\Delta$ | = | $167145119796953088$ | = | $2^{20} \cdot 3^{16} \cdot 7 \cdot 23^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{54804145548726848737}{637608031452} \) | = | $2^{-2} \cdot 3^{-16} \cdot 7^{-1} \cdot 23^{-2} \cdot 193^{3} \cdot 19681^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4556953238797556755508397642$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4159745530398377114249915820$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0156853826442096$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.602815409495771$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $8.2519610819874383350267468160$ |
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| Real period: | $\Omega$ | ≈ | $0.29262139638798820340198957725$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.8294007495009964296057731313 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.829400750 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.292621 \cdot 8.251961 \cdot 8}{2^2} \\ & \approx 4.829400750\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 786432 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}^{*}$ | additive | -1 | 6 | 20 | 2 |
| $3$ | $2$ | $I_{16}$ | nonsplit multiplicative | 1 | 1 | 16 | 16 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $23$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 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 |
|---|---|---|---|
| $2$ | 2B | 16.48.0.194 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2576 = 2^{4} \cdot 7 \cdot 23 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 2478 & 2563 \end{array}\right),\left(\begin{array}{rr} 2563 & 2560 \\ 2188 & 2247 \end{array}\right),\left(\begin{array}{rr} 384 & 5 \\ 691 & 2562 \end{array}\right),\left(\begin{array}{rr} 2561 & 16 \\ 2560 & 17 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 2572 & 2573 \end{array}\right),\left(\begin{array}{rr} 314 & 1927 \\ 2251 & 1286 \end{array}\right),\left(\begin{array}{rr} 2365 & 16 \\ 192 & 2261 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[2576])$ is a degree-$68942168064$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2576\Z)$.
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 | $2$ | \( 7 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 10304 = 2^{6} \cdot 7 \cdot 23 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 4416 = 2^{6} \cdot 3 \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 1344 = 2^{6} \cdot 3 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 30912bk
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 966g5, its twist by $-8$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{14}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{7})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{23})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{161})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.2157642632986624.35 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1927561216.6 | \(\Z/8\Z\) | not in database |
| $8$ | \(\Q(\sqrt{2}, \sqrt{7}, \sqrt{23})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | 16.0.59447875862838378496.4 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | nonsplit | ord | nonsplit | ord | ord | ord | ord | split | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 0 | 0 | 0 | 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.