Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-87588545x+315619808449\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-87588545xz^2+315619808449z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7094672172x+230065556342832\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{8377140904}{1550025}, \frac{16856287441433}{1929781125}\right) \) | $15.566497139124867479425305732$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([10429540425480:16856287441433:1929781125]\) | $15.566497139124867479425305732$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{8376624229}{172225}, \frac{16856287441433}{71473375}\right) \) | $15.566497139124867479425305732$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 448448 \) | = | $2^{6} \cdot 7^{2} \cdot 11 \cdot 13$ |
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| Minimal Discriminant: | $\Delta$ | = | $-20798325060785739923456$ | = | $-1 \cdot 2^{19} \cdot 7^{6} \cdot 11^{10} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{2409558590804994721}{674373039626} \) | = | $-1 \cdot 2^{-1} \cdot 11^{-10} \cdot 13^{-1} \cdot 29^{3} \cdot 46229^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2639312588575538906617249290$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2512554134899792739832003751$ |
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| $abc$ quality: | $Q$ | ≈ | $1.104756801261857$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.108412312868199$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $15.566497139124867479425305732$ |
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| Real period: | $\Omega$ | ≈ | $0.11851269838448695331517318615$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot1\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.3793103214083378507625294235 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.379310321 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.118513 \cdot 15.566497 \cdot 4}{1^2} \\ & \approx 7.379310321\end{aligned}$$
Modular invariants
Modular form 448448.2.a.bq
For more coefficients, see the Downloads section to the right.
| Modular degree: | 38016000 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{9}^{*}$ | additive | 1 | 6 | 19 | 1 |
| $7$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $13$ | $1$ | $I_{1}$ | split 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 | $\ell$-adic index |
|---|---|---|---|
| $5$ | 5B.4.2 | 5.12.0.2 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 3119 & 0 \\ 0 & 3639 \end{array}\right),\left(\begin{array}{rr} 911 & 1050 \\ 1435 & 1611 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 3585 & 3521 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1177 & 2870 \\ 3080 & 281 \end{array}\right),\left(\begin{array}{rr} 1819 & 2590 \\ 1295 & 2029 \end{array}\right),\left(\begin{array}{rr} 561 & 1050 \\ 3325 & 1611 \end{array}\right),\left(\begin{array}{rr} 3631 & 10 \\ 3630 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3640])$ is a degree-$811550638080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3640\Z)$.
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 | $4$ | \( 637 = 7^{2} \cdot 13 \) |
| $5$ | good | $2$ | \( 40768 = 2^{6} \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 9152 = 2^{6} \cdot 11 \cdot 13 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 40768 = 2^{6} \cdot 7^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 34496 = 2^{6} \cdot 7^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 448448bq
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 286d2, its twist by $-56$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.