Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-7483x+245833\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-7483xz^2+245833z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-9698643x+11615060718\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(48, -35)$ | $0.83743399821762114393700516678$ | $\infty$ |
| $(81, 383)$ | $1.7241732960950729176305175493$ | $\infty$ |
| $(203/4, -203/8)$ | $0$ | $2$ |
Integral points
\( \left(-84, 581\right) \), \( \left(-84, -497\right) \), \( \left(26, 251\right) \), \( \left(26, -277\right) \), \( \left(48, -13\right) \), \( \left(48, -35\right) \), \( \left(57, 64\right) \), \( \left(57, -121\right) \), \( \left(63, 140\right) \), \( \left(63, -203\right) \), \( \left(81, 383\right) \), \( \left(81, -464\right) \), \( \left(323, 5465\right) \), \( \left(323, -5788\right) \)
Invariants
| Conductor: | $N$ | = | \( 53130 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $48511196580$ | = | $2^{2} \cdot 3 \cdot 5 \cdot 7^{4} \cdot 11^{4} \cdot 23 $ |
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| j-invariant: | $j$ | = | \( \frac{46349134440566329}{48511196580} \) | = | $2^{-2} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-4} \cdot 11^{-4} \cdot 23^{-1} \cdot 359209^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.96769864054258486203089991914$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.96769864054258486203089991914$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8953827865765887$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.5269509236338394$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.2731035552396061531230408698$ |
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| Real period: | $\Omega$ | ≈ | $1.1249346108448391566855393374$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot1\cdot1\cdot2\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $5.7286330099145901535962362828 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.728633010 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.124935 \cdot 1.273104 \cdot 16}{2^2} \\ & \approx 5.728633010\end{aligned}$$
Modular invariants
Modular form 53130.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 90112 |
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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 semistable. There are 6 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $11$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $23$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 212520 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 212514 & 212515 \end{array}\right),\left(\begin{array}{rr} 70844 & 1 \\ 141703 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 60721 & 8 \\ 30364 & 33 \end{array}\right),\left(\begin{array}{rr} 85016 & 3 \\ 5 & 2 \end{array}\right),\left(\begin{array}{rr} 212513 & 8 \\ 212512 & 9 \end{array}\right),\left(\begin{array}{rr} 57961 & 8 \\ 19324 & 33 \end{array}\right),\left(\begin{array}{rr} 79699 & 79698 \\ 26578 & 132835 \end{array}\right),\left(\begin{array}{rr} 92408 & 3 \\ 110885 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 185963 & 185956 \\ 186002 & 79701 \end{array}\right)$.
The torsion field $K:=\Q(E[212520])$ is a degree-$5241810922242048000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/212520\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 345 = 3 \cdot 5 \cdot 23 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 17710 = 2 \cdot 5 \cdot 7 \cdot 11 \cdot 23 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 10626 = 2 \cdot 3 \cdot 7 \cdot 11 \cdot 23 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 7590 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 23 \) |
| $11$ | split multiplicative | $12$ | \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \) |
| $23$ | nonsplit multiplicative | $24$ | \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 53130.d
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{345}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{115}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{115})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | 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 |
| $16$ | 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 | nonsplit | nonsplit | nonsplit | nonsplit | split | ord | ord | ss | nonsplit | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 5 | 2 | 2 | 2 | 3 | 2 | 2 | 2,4 | 2 | 2 | 2,2 | 2 | 4 | 2 | 2,2 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0,0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.