Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-114867x+20411041\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-114867xz^2+20411041z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1837875x+1304468750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-156, 5953)$ | $3.4342468416749404189701418612$ | $\infty$ |
| $(-406, 203)$ | $0$ | $2$ |
Integral points
\( \left(-406, 203\right) \), \( \left(-156, 5953\right) \), \( \left(-156, -5797\right) \), \( \left(1115, 35186\right) \), \( \left(1115, -36301\right) \)
Invariants
| Conductor: | $N$ | = | \( 76050 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-82470556898437500$ | = | $-1 \cdot 2^{2} \cdot 3^{7} \cdot 5^{9} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{24389}{12} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-1} \cdot 29^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9512285966677425506070870100$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0876306607226559440678488292$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1033940611435318$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.198139390180232$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.4342468416749404189701418612$ |
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| Real period: | $\Omega$ | ≈ | $0.31874266687067975124406038973$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.3785639880307184014157486431 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.378563988 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.318743 \cdot 3.434247 \cdot 16}{2^2} \\ & \approx 4.378563988\end{aligned}$$
Modular invariants
Modular form 76050.2.a.y
For more coefficients, see the Downloads section to the right.
| Modular degree: | 614400 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $13$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $5$ | 5B.4.2 | 5.12.0.2 |
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 $288$, genus $5$, and generators
$\left(\begin{array}{rr} 781 & 1326 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1338 & 715 \\ 65 & 194 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 391 & 1326 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 714 & 1547 \\ 845 & 64 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 1320 & 1211 \end{array}\right),\left(\begin{array}{rr} 1541 & 20 \\ 1540 & 21 \end{array}\right),\left(\begin{array}{rr} 479 & 0 \\ 0 & 1559 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$3220439040$ 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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \) |
| $3$ | additive | $8$ | \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \) |
| $5$ | additive | $14$ | \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 76050cq
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 150a1, its twist by $-195$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.4056000.2 | \(\Z/4\Z\) | not in database |
| $4$ | 4.4.190125.1 | \(\Z/10\Z\) | not in database |
| $8$ | 8.0.333135504000000.16 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.148060224000000.54 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.36147515625.2 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $10$ | 10.0.146163202380000000.1 | \(\Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/20\Z\) | not in database |
| $20$ | 20.0.21363681729976837664400000000000000.2 | \(\Z/2\Z \oplus \Z/10\Z\) | 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 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | add | add | ord | ord | add | ord | ss | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | - | - | 1 | 1 | - | 1 | 1,1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | - | - | 0 | 0 | - | 0 | 0,0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.