Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+263290677x-17576285113478\)
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(homogenize, simplify) |
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\(y^2z=x^3+263290677xz^2-17576285113478z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+263290677x-17576285113478\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(22646, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([22646:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(22646, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(22646, 0\right) \)
\([22646:0:1]\)
\( \left(22646, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 121680 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 13^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-134624062101515580112896000000$ | = | $-1 \cdot 2^{21} \cdot 3^{18} \cdot 5^{6} \cdot 13^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{63745936931123}{4251528000000} \) | = | $2^{-9} \cdot 3^{-12} \cdot 5^{-6} \cdot 43^{3} \cdot 929^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.2689434745180360285608774383$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1027781315278833214059071172$ |
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| $abc$ quality: | $Q$ | ≈ | $1.089200510549715$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.364107501216725$ | |||
| 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: | $\Omega$ | ≈ | $0.015652052390565049762239601645$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.1304104781130099524479203290 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $25$ = $5^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.130410478 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{25 \cdot 0.015652 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 3.130410478\end{aligned}$$
Modular invariants
Modular form 121680.2.a.bn
For more coefficients, see the Downloads section to the right.
| Modular degree: | 129392640 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{13}^{*}$ | additive | -1 | 4 | 21 | 9 |
| $3$ | $4$ | $I_{12}^{*}$ | additive | -1 | 2 | 18 | 12 |
| $5$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $13$ | $2$ | $III^{*}$ | 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 | 2.3.0.1 | $3$ |
| $3$ | 3Ns | 3.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $144$, genus $5$, and generators
$\left(\begin{array}{rr} 10 & 3 \\ 753 & 1552 \end{array}\right),\left(\begin{array}{rr} 7 & 12 \\ 144 & 247 \end{array}\right),\left(\begin{array}{rr} 937 & 12 \\ 942 & 73 \end{array}\right),\left(\begin{array}{rr} 326 & 207 \\ 195 & 1106 \end{array}\right),\left(\begin{array}{rr} 492 & 5 \\ 187 & 1488 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 5 & 12 \\ 1512 & 1445 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1549 & 12 \\ 1548 & 13 \end{array}\right),\left(\begin{array}{rr} 783 & 1042 \\ 22 & 795 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$6440878080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $4$ | \( 117 = 3^{2} \cdot 13 \) |
| $3$ | additive | $2$ | \( 2704 = 2^{4} \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 24336 = 2^{4} \cdot 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $62$ | \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 121680.bn
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 5070.l2, its twist by $156$.
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 |
| $4$ | \(\Q(\sqrt{-78 +18 \sqrt{65}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{26 +10 \sqrt{13}})\) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-26 +2 \sqrt{-39}})\) | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1000887114240000.47 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.100088711424.4 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.2846967791616.3 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.25622710124544.12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/6\Z \oplus \Z/6\Z\) | 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 | 13 |
|---|---|---|---|---|
| Reduction type | add | add | nonsplit | add |
| $\lambda$-invariant(s) | - | - | 4 | - |
| $\mu$-invariant(s) | - | - | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ 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 $0$.