Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-23526208x+43926990912\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-23526208xz^2+43926990912z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1905622875x+32017059506250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(2817, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([2817:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(25350, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(2817, 0\right) \)
\([2817:0:1]\)
\( \left(2817, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 67600 \) | = | $2^{4} \cdot 5^{2} \cdot 13^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $84835994984000000000$ | = | $2^{12} \cdot 5^{9} \cdot 13^{9} $ |
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| j-invariant: | $j$ | = | \( 16974593 \) | = | $257^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8861440369780718092437964338$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.93779359600360133316412076875$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9377878821365829$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.622914958992652$ | |||
| 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: | $\Omega$ | ≈ | $0.18162433101557352572763788483$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.9059892962491764116422061572 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.905989296 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.181624 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 2.905989296\end{aligned}$$
Modular invariants
Modular form 67600.2.a.cy
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3194880 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | -1 | 4 | 12 | 0 |
| $5$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $13$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
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 | 16.24.0.22 | $24$ |
| $3$ | 3Nn | 3.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \), index $1152$, genus $41$, and generators
$\left(\begin{array}{rr} 2117 & 48 \\ 660 & 1313 \end{array}\right),\left(\begin{array}{rr} 3073 & 48 \\ 3072 & 49 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 24 & 577 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 48 & 1 \end{array}\right),\left(\begin{array}{rr} 1424 & 3103 \\ 1 & 256 \end{array}\right),\left(\begin{array}{rr} 33 & 16 \\ 2000 & 2577 \end{array}\right),\left(\begin{array}{rr} 1 & 48 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 19 & 18 \\ 210 & 199 \end{array}\right),\left(\begin{array}{rr} 1393 & 3072 \\ 666 & 2431 \end{array}\right),\left(\begin{array}{rr} 2081 & 48 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1216 & 3093 \\ 1203 & 976 \end{array}\right)$.
The torsion field $K:=\Q(E[3120])$ is a degree-$12881756160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3120\Z)$.
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 | $2$ | \( 65 = 5 \cdot 13 \) |
| $5$ | additive | $14$ | \( 2704 = 2^{4} \cdot 13^{2} \) |
| $13$ | additive | $62$ | \( 400 = 2^{4} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 67600.cy
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 4225.h1, its twist by $-260$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{65}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.4.274625.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.19307236000000.17 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.521295372000000.4 | \(\Z/6\Z\) | 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 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 5 | 13 |
|---|---|---|---|
| Reduction type | add | add | add |
| $\lambda$-invariant(s) | - | - | - |
| $\mu$-invariant(s) | - | - | - |
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 $0$.