Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-1153795x-476736893\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-1153795xz^2-476736893z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-18460715x-30529621850\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 414050 \) | = | $2 \cdot 5^{2} \cdot 7^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-242189968384000$ | = | $-1 \cdot 2^{13} \cdot 5^{3} \cdot 7^{2} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{5745702166029}{8192} \) | = | $-1 \cdot 2^{-13} \cdot 3^{3} \cdot 7 \cdot 3121^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0315072129542402848010293129$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.022354697939061272273203634902$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0876250170823247$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.135640390029051$ | |||
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: | $\Omega$ | ≈ | $0.072834541075913224660714054978$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 26 $ = $ 13\cdot2\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.8936980679737438411785654294 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.893698068 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.072835 \cdot 1.000000 \cdot 26}{1^2} \\ & \approx 1.893698068\end{aligned}$$
Modular invariants
Modular form 414050.2.a.fq
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3841344 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $13$ | $I_{13}$ | split multiplicative | -1 | 1 | 13 | 13 |
| $5$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $7$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
| $13$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 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 |
|---|---|---|
| $13$ | 13B | 13.14.0.1 |
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 $336$, genus $9$, and generators
$\left(\begin{array}{rr} 1063 & 26 \\ 2080 & 3527 \end{array}\right),\left(\begin{array}{rr} 1834 & 13 \\ 2717 & 3628 \end{array}\right),\left(\begin{array}{rr} 895 & 1794 \\ 416 & 1351 \end{array}\right),\left(\begin{array}{rr} 1 & 26 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 14 & 23 \\ 871 & 1431 \end{array}\right),\left(\begin{array}{rr} 14 & 13 \\ 1807 & 3628 \end{array}\right),\left(\begin{array}{rr} 746 & 13 \\ 1287 & 3518 \end{array}\right),\left(\begin{array}{rr} 3615 & 26 \\ 3614 & 27 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 26 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3640])$ is a degree-$115935805440$ 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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 41405 = 5 \cdot 7^{2} \cdot 13^{2} \) |
| $5$ | additive | $10$ | \( 16562 = 2 \cdot 7^{2} \cdot 13^{2} \) |
| $7$ | additive | $14$ | \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
13.
Its isogeny class 414050fq
consists of 2 curves linked by isogenies of
degree 13.
Twists
The minimal quadratic twist of this elliptic curve is 2450o2, its twist by $13$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.