Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-72500276x+208820340880\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-72500276xz^2+208820340880z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-93960357723x+9743003705170422\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 46090 \) | = | $2 \cdot 5 \cdot 11 \cdot 419$ |
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| Minimal Discriminant: | $\Delta$ | = | $5550424680603960608000000$ | = | $2^{11} \cdot 5^{6} \cdot 11^{9} \cdot 419^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{42144673353901398176116477249}{5550424680603960608000000} \) | = | $2^{-11} \cdot 5^{-6} \cdot 11^{-9} \cdot 241^{3} \cdot 419^{-3} \cdot 3457^{3} \cdot 4177^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.4760388941046278678352000129$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.4760388941046278678352000129$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9891540990392442$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.137898086963403$ | |||
| 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: | $\Omega$ | ≈ | $0.073322324324382431298422483851$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 22 $ = $ 11\cdot2\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.6130911351364134885652946447 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.613091135 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.073322 \cdot 1.000000 \cdot 22}{1^2} \\ & \approx 1.613091135\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 13514688 |
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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 semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $11$ | $I_{11}$ | split multiplicative | -1 | 1 | 11 | 11 |
| $5$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $11$ | $1$ | $I_{9}$ | nonsplit multiplicative | 1 | 1 | 9 | 9 |
| $419$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 |
|---|---|---|---|
| $3$ | 3B.1.2 | 3.8.0.2 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 110616 = 2^{3} \cdot 3 \cdot 11 \cdot 419 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 106010 & 4611 \\ 96799 & 55315 \end{array}\right),\left(\begin{array}{rr} 55309 & 6 \\ 55311 & 19 \end{array}\right),\left(\begin{array}{rr} 27655 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 60337 & 6 \\ 70395 & 19 \end{array}\right),\left(\begin{array}{rr} 110611 & 6 \\ 110610 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 45673 & 6 \\ 26403 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[110616])$ is a degree-$1870261261811712000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/110616\Z)$.
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 | $4$ | \( 4609 = 11 \cdot 419 \) |
| $3$ | good | $2$ | \( 2 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 9218 = 2 \cdot 11 \cdot 419 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 2095 = 5 \cdot 419 \) |
| $419$ | nonsplit multiplicative | $420$ | \( 110 = 2 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 46090g
consists of 2 curves linked by isogenies of
degree 3.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.3.36872.1 | \(\Z/2\Z\) | not in database |
| $3$ | \(\Q(\sqrt[3]{2})\) | \(\Z/3\Z\) | not in database |
| $6$ | 6.6.50129120526848.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.34992.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.36707698368.2 | \(\Z/6\Z\) | not in database |
| $9$ | 9.3.15787063669279186944.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.13954753530733668321469687921946317984942456687500000000.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.6729247241042620293415107551385901596672.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.3123437462973934150480218079643450098891868268920832.1 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 3 | 5 | 7 | 11 | 419 |
|---|---|---|---|---|---|---|
| Reduction type | split | ord | nonsplit | ord | nonsplit | nonsplit |
| $\lambda$-invariant(s) | 5 | 2 | 0 | 0 | 2 | 0 |
| $\mu$-invariant(s) | 0 | 1 | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 13$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.