Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-15684247x-23908018986\)
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(homogenize, simplify) |
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\(y^2z=x^3-15684247xz^2-23908018986z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-15684247x-23908018986\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(10123559137/178084, 1016049732761415/75151448)$ | $18.346927619706410754806542093$ | $\infty$ |
| $(-2286, 0)$ | $0$ | $2$ |
| $(4573, 0)$ | $0$ | $2$ |
Integral points
\( \left(-2287, 0\right) \), \( \left(-2286, 0\right) \), \( \left(4573, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 10640 \) | = | $2^{4} \cdot 5 \cdot 7 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $35423365464121600$ | = | $2^{8} \cdot 5^{2} \cdot 7^{6} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{1666766511378391624080336}{138372521344225} \) | = | $2^{4} \cdot 3^{3} \cdot 5^{-2} \cdot 7^{-6} \cdot 19^{-6} \cdot 523^{3} \cdot 29989^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6185697930687676043270326451$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.1564716726954707313822112308$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0836700714853877$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.612987520469785$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $18.346927619706410754806542093$ |
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| Real period: | $\Omega$ | ≈ | $0.075863633344345330446686395563$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2\cdot2\cdot2\cdot( 2 \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.1755937698099487014174139338 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.175593770 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.075864 \cdot 18.346928 \cdot 48}{4^2} \\ & \approx 4.175593770\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 276480 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | 1 | 4 | 8 | 0 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $19$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
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 |
|---|---|---|
| $2$ | 2Cs | 4.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2660 = 2^{2} \cdot 5 \cdot 7 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 2657 & 4 \\ 2656 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1541 & 4 \\ 422 & 9 \end{array}\right),\left(\begin{array}{rr} 1329 & 2658 \\ 0 & 2659 \end{array}\right),\left(\begin{array}{rr} 381 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1597 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[2660])$ is a degree-$238281523200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2660\Z)$.
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 | $2$ | \( 1 \) |
| $3$ | good | $2$ | \( 80 = 2^{4} \cdot 5 \) |
| $5$ | split multiplicative | $6$ | \( 2128 = 2^{4} \cdot 7 \cdot 19 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 1520 = 2^{4} \cdot 5 \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 560 = 2^{4} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 10640f
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 5320e2, its twist by $-4$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-19}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{35})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{19}, \sqrt{35})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.50064115360000.72 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.5598720000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | ss | split | nonsplit | ord | ord | ord | split | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3,3 | 4 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 3 |
| $\mu$-invariant(s) | - | 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.