Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2+16305x-734193\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z+16305xz^2-734193z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+21131253x-34571469114\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(423/4, 11403/8)$ | $2.2549151150334715652682428632$ | $\infty$ |
| $(163/4, -167/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 5070 \) | = | $2 \cdot 3 \cdot 5 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-514616275046250$ | = | $-1 \cdot 2 \cdot 3^{8} \cdot 5^{4} \cdot 13^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{99317171591}{106616250} \) | = | $2^{-1} \cdot 3^{-8} \cdot 5^{-4} \cdot 11^{3} \cdot 13^{-1} \cdot 421^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5095072215795234220066089557$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.22703254284875505397986523492$ |
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| $abc$ quality: | $Q$ | ≈ | $1.035787691267508$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.772104311485498$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.2549151150334715652682428632$ |
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| Real period: | $\Omega$ | ≈ | $0.28237707753676440655091479270$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 1\cdot2\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.0938907222130290504535221492 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.093890722 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.282377 \cdot 2.254915 \cdot 32}{2^2} \\ & \approx 5.093890722\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 21504 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $13$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
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 | 4.12.0.8 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 520 = 2^{3} \cdot 5 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 328 & 73 \\ 353 & 400 \end{array}\right),\left(\begin{array}{rr} 513 & 8 \\ 512 & 9 \end{array}\right),\left(\begin{array}{rr} 356 & 519 \\ 97 & 514 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 456 & 203 \\ 329 & 358 \end{array}\right),\left(\begin{array}{rr} 417 & 8 \\ 108 & 33 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 514 & 515 \end{array}\right)$.
The torsion field $K:=\Q(E[520])$ is a degree-$402554880$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/520\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 169 = 13^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 1690 = 2 \cdot 5 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 5070q
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 390a4, its twist by $13$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-26}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{26}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | not in database |
| $4$ | 4.0.4499456.4 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{26})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.80980417183744.23 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.74870947840000.36 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.12356631040000.59 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 |
| $16$ | 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 | split | nonsplit | split | ss | ss | add | ord | ss | ord | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 2 | 1 | 4 | 1,1 | 1,1 | - | 1 | 1,1 | 1 | 1 | 1,3 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 2 | 0 | 0 | 0,0 | 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.